natural structures in differential geometry lucas...
TRANSCRIPT
NATURAL STRUCTURES IN DIFFERENTIALGEOMETRY
Lucas Mason-Brown
A thesis submitted in partial fulfillment of therequirements for the degree of Master of Science in
mathematics
Trinity College Dublin
March 2015
Declaration
I declare that the thesis contained herein is entirely my own work, ex-cept where explicitly stated otherwise, and has not been submitted for adegree at this or any other university. I grant Trinity College Dublin per-mission to lend or copy this thesis upon request, with the understandingthat this permission covers only single copies made for study purposes,subject to normal conditions of acknowledgement.
Contents
1 Summary 3
2 Acknowledgment 6
I Natural Bundles and Operators 6
3 Jets 6
4 Natural Bundles 9
5 Natural Operators 11
6 Invariant-Theoretic Reduction 16
7 Classical Results 24
8 Natural Operators on Differential Forms 32
II Natural Operators on Alternating Multivector Fields 38
9 Twisted Algebras in VecZ 40
10 Ordered Multigraphs 48
11 Ordered Multigraphs and Natural Operators 52
12 Gerstenhaber Algebras 57
13 Pre-Gerstenhaber Algebras 58
14 A Lie-admissable Structure on Grairred
acyc 63
15 A Co-algebra Structure on Grairred
acyc 67
16 Loday’s Rigidity Theorem 75
17 A Useful Consequence of Kunneth’s Theorem 83
18 Chevalley-Eilenberg Cohomology 87
1 Summary
Many of the most important constructions in differential geometry are functorial. Take, for
example, the tangent bundle. The tangent bundle can be profitably viewed as a functor T :
Manm → Fib from the category of m-dimensional manifolds (with local diffeomorphisms) to
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the category of fiber bundles (with locally invertible bundle maps). This functor takes every
m-dimensional manifold M to its tangent bundle TM := tm∈MTmM and every local diffeomor-
phism f : M → N to its pushforward Tf := f∗ : TM → TN . The statement that T is a functor
is essentially the chain rule.
Functors of the general type Manm → Fib are ubiquitous in differential geometry. Other
examples include: the cotangent bundle T ∗, tensor bundles and their various sub-bundles (most
notably the exterior bundles ∧nT and ∧nT ∗), frame bundles, jet bundles, and others. Over
the years such functors have been variously termed “bundle functors,” “natural bundles,” and
“geometric objects.” In this thesis, we use the term “natural bundle” throughout.
If we view natural bundles as functors, certain constructions play the role of natural transfor-
mations. Other constructions play the role of natural operators, transforming smooth sections
of one bundle into smooth sections of another, through a rule or recipe which respects the local
action of diffeomorphisms. Classical examples include: the exterior derivative, the wedge prod-
uct, the contraction (i.e. the evaluation of a form at a vector field), the Lie bracket, and the
much-more-general Schouten-Nijenhuis bracket of alternating multivector fields.
The functorial approach to natural bundles and operators was first introduced by A. Nijenhuis
in his talk at the 1958 International Congress of Mathematicians. The foundations of a general
theory were later developed by C.L. Terng and R. Palais ([1], [2]) with important contributions
from D.B.A. Epstein, W.P. Thurston ([3]), and others. Following in the early footsteps of R.
Palais, who demonstrated in 1959 that the exterior derivative is, apart from the identity, the only
unary natural operator on differential forms ([4]), recent work in the field of “natural differential
geometry” has largely been devoted to the task of classifying natural bundles and operators
of a prescribed type. Much of this work is summarized in the foundational monograph of I.
Kolar, P. Michor, and J. Slovak ([5]), which was the primary reference for the first part of this
thesis. However, since the publication of this monograph, a few major holes have been filled.
Over the past decade, M. Markl has published a string of papers, culminating in an exhaustive
classification of natural n-ary operators on vector fields. He demonstrates, not surprisingly, that
over high-dimensional manifolds the only such operators are given by linear combinations of
various iterations of the Lie bracket. His work draws heavily on Operad theory, particularly on
the work of F. Chapoton, M. Livernet ([6]), and J.-L Loday ([7]), as well as a graphical approach
to tensor calculus originally formalized by S.A. Merkulov in [8].
We see in the foundational work of R. Palais and the more recent work of M. Markl the contours
of a much more general problem. Namely, what is the complete space of natural multilinear
operators on differential forms and alternating multivector fields? More explicitly, if ∧αiT ∗ is
the natural bundle of differential forms of degree αi and ∧βiT is the natural bundle of alternating
multivector fields of degree βi, can we classify the space of natural multilinear operators of the
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general type:
η : ∧α1T ∗ × ...× ∧αpT ∗ × ∧β1T × ...× ∧βqT ∧α′1T ∗ × ...× ∧α
′r × ∧β
′1T × ...× ∧β
′sT (1)
We conjecture that all such operators can be built up from 5 ‘elementary’ operations. On
differential forms, there is the exterior derivative and the wedge product of differential forms.
On alternating multivector fields, there is the Schouten-Nijenhuis bracket and the wedge product
of alternating multivector fields. Linking the two is the contraction. We believe that (over high-
dimensional manifolds) all natural operators of type (1) can be realized as linear combinations of
iterations of these 5 operations. A proof of this conjecture would have intrinsic value within the
field of natural differential geometry. Such a proof would also be valuable in a much broader sense.
Natural operators are precisely the kinds of operators that present themselves in mathematical
formulations of fundamental physics. Furthermore, differential forms and alternating multivector
fields are among the most pervasive mathematical objects in these fundamental laws. Thus, a
proof of this conjecture would radically limit the range of plausible natural laws that physicists
might be tempted to formulate. In this thesis, we make several steps towards proving this
conjecture.
In the first part, we develop the basic theory of natural differential operators, roughly following
chapters IV, V, and VI of [5]. The key concept in the first part is the method of invariant-
theoretic reduction, whereby the geometric problem of classifying natural differential operators is
reduced to the much more tractable algebraic problem of classifying module homomorphisms of
a particular source and target. We conclude part I with an original proof that all natural n-ary
operators on differential forms are exhausted by iterations of the wedge product and exterior
derivative (making no assumption of multilinearity).
In the second and final part, we shift our attention to the more obstreperous case of alternating
multivector fields. Generalizing the work of M. Markl, we introduce the notion of an ordered
multigraph as a general framework for the study of natural operators on alternating multivec-
tor fields. Inspired by Markl, we draw connections between the space of natural operators on
alternating multivector fields and the cohomology of a particular complex of multigraphs. We
then develop the notion of a pre-Gerstenhaber algebra, which combines the structure of a graded
commutative algebra with the structure of a graded Lie-admissable algebra, and prove that these
operads are compatible in a suitable sense. We immediately see that this is the correct algebraic
structure for the geometric problem at hand. Finally, we apply Loday’s rigidity theorem to
deduce certain facts about the structure of the relevant algebras.
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2 Acknowledgment
Special thanks go to Vladimir Dotsenko. Without his patient support and insight, none of this
would have been possible. Thanks also to my undergraduate mentors and advisors—Benedict
Gross, Mike Rosen, Dan Abramovic, and Tom Goodwillie, just to name a few—who enabled and
empowered me as a mathematician.
Part I
Natural Bundles and Operators
3 Jets
Suppose f : R→ R is a smooth function and x0 ∈ R. By Taylor’s theorem
f(x) = f(x0) + f ′(x0)(x− x0) + ...+f (r)(x0)
r!(x− x0)r + hr(x)(x− x0)r
where limx→x0hr(x) = 0.
Definition 1. The r-jet of f : R→ R is the rth order Taylor polynomial
jrx0(x) = f(x0) + f ′(x0)x+ ...+
f (r)(x0)
r!xr
How can we extend this notion to smooth maps between arbitrary smooth manifolds? One idea
is to pick chart maps near x and f(x) and consider multivariate Taylor polynomials in these
chosen local coordinates. We wish to formalize this approach in a coordinate-free manner. We
begin by defining jets of smooth curves. Let M be a manifold and x ∈ M . By a smooth curve
through x we mean a smooth map f : R → M with f(0) = x. Two smooth curves f and g
through x are equivalent to order r if for every smooth function ψ : M → R, we have
jr0(ψ ◦ f) = jr0(ψ ◦ g)
Note, these expressions make sense, since ψ ◦ f and ψ ◦ g are smooth functions from R to itself.
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Definition 2. The r-jet of a smooth curve f through x is its equivalence class under the equiv-
alence relation defined above.
At the risk of abusing notation, we will denote the r-jet of f by jr0(f). Two smooth maps
f, g : M → N are equivalent to order r at x if for every smooth curve φ : R→M through x, we
have
jr0(f ◦ φ) = jr0(g ◦ φ)
Definition 3. The r-jet of a smooth map f : M → N at x is its equivalence class under the
equivalence relation defined above.
We will denote the r-jet of f : M → N at x by jrx(f). The set of all r-jets is denoted by
Jr(M,N). The set of all r-jets at x is Jrx(M,N). Finally, the set of all r-jets at x with f(x) = y
is Jrx(M,N)y.
Next, we show that jet composition makes sense.
Proposition 1. Suppose f, f ′ : M → N and g, g′ : N → O such that f(x) = y = f ′(x). If
jrx(f) = jrx(f ′) and jry(g) = jry(g′), then jrx(g ◦ f) = jrx(g′ ◦ f ′).
Proof. First note that if two curves a, b : R→M are rth order equivalent at x, then jrx(f ◦ a) =
jrx(f ◦b) for any smooth map f : M → N . Indeed, for every function ψ : N → R, the composition
ψ ◦ f is a function on M .
Take any curve φ : R → M with φ(0) = x. jrx(f) = jrx(f ′) implies jrxf ◦ φ = jrxf′ ◦ φ. And thus
the remark above gives jrx(g′ ◦ f ◦ φ) = jrx(g′ ◦ f ′ ◦ φ). On the other hand, jry(g) = jry(g′) yields
jrx(g ◦ f ◦ φ) = jrx(g′ ◦ f ◦ φ). Thus, jrx(g ◦ f ◦ φ) = jrx(g′ ◦ f ′ ◦ φ) and so jrx(g ◦ f) = jrx(g′ ◦ f ′) by
definition.
Thus, r-jets are the morphisms of a category with smooth manifolds as objects. The set of
invertible r-jets between manifolds M and N is denoted invJr(M,N). Note that an r-jet is
invertible if and only if the underlying 1-jet is invertible (an immediate consequence of the
inverse function theorem). Such a jet exists between manifolds M and N if and only if dimM =
dimN .
One can also consider jets of infinite order. Jets of finite order form a projective system
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J1(M,N)π21← J2(M,N)
π32← J3(M,N)
π43← ...
We define J∞(M,N) as its inverse limit. Thus, a jet of infinite order is an infinite sequence of
finite order jets j1, j2, j3, ... such that ji = πi+1i (ji+1) for all i ∈ N.
Next we define a special group of invertible jets.
Definition 4. The rth jet group in dimension m is the set Grm := invJr0 (Rm,Rm)0. Grm forms
a group under jet composition.
In fact, Grm is a Lie group. Its manifold structure is determined by the inclusion
Grm ⊂ {m-tuples of degree r polynomials in m variables} = Rm(m+rm )
Note that G1m is canonically identified with GLm(R). The kernel of the jet projection Grm →
GLm(R) is a (connected, simply connected) normal subgroup which we will denote by NGLrm.
Thus, we have a short exact sequence
1→ NGLrm → Grm → GLm(R)→ 1
which splits on the right through the obvious inclusion GLm(R) ↪→ Grm.
Remark 1. It follows from these observations and the splitting lemma for (non-abelian) groups
that Grm is the semidirect product NGLrmoGLm(R) where GLm(R) acts on NGLrm by adjunc-
tion. This decomposition will prove particularly useful in part 2.
The Lie algebra of Grm is the space of r-jets of vector fields on Rm that vanish at the origin, or,
equivalently, vector fields on Rm with polynomial coefficients of degree not exceeding r and no
constant terms. As a vector space
grm = {a1 + a2 + ...+ ar | ai ∈ Sym(⊗iRm,Rm
)}
The Lie algebra of NGLrm is the (nilpotent) ideal:
nglrm = {a2 + a3...+ ar | ai ∈ Sym(⊗iRm,Rm
)}
For a thorough treatment of jet groups and their Lie algebras, see [5, chap. 4].
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Finally, we define rth order velocities on M .
Definition 5. T rk := Jr0 (Rk,M) is the set of k-dimensional velocities of order r on M .
Picking charts on M , T rk is seen to be a manifold (in fact, a fibered manifold over M).
4 Natural Bundles
A natural bundle, in the sense of [5], generalizes tangent and cotangent bundles in differential ge-
ometry. These objects provide recipes for constructing fiber bundles over arbitrarym-dimensional
manifolds—recipes which respect the action of local diffeomorphisms. For example, the tangent
bundle M 7→ TM comes equipped with the pushforward map f 7→ f∗ which satisfies the chain
rule (f ◦ g)∗ = f∗ ◦ g∗ on compositions.
Let Manm be the category of m-dimensional manifolds, where morphisms are local diffeomor-
phisms (i.e. globally defined maps with locally defined inverses) and let Fib be the category of
fiber bundles, where morphisms are locally invertible bundle maps.
Definition 6. A natural bundle over m-dimensional manifolds is a (covariant) functor F :
Manm → Fib satisfying
1. (Prolongation) B ◦ F = IdManm , where B : Fib → Manm is the base functor. In other
words, FM is a bundle over M for every manifold M , and Ff covers f for every local
diffeomorphism f : M → N .
2. (Locality) If i : U → M is an inclusion of an embedded submanifold, then FU = π−1M (U)
and Fi is the inclusion of π−1M (U) into FM .
A natural vector bundle is a natural bundle with values in the subcategory of vector bundles.
Example 1. Perhaps the most important and ubiquitous examples of natural bundles in dif-
ferential geometry are the tensor bundles (i.e. tangent bundles, cotangent bundles, and their
various tensor products). The tangent bundle acts on morphisms via the pushforward map, and
the cotangent bundle acts on morphisms by pulling back along their (local) inverses. Higher
tensor bundles act via tensor products of these actions.
Example 2. For every manifold F , there is a (trivial) natural bundle ×F : M 7→ M × F . On
morphisms, ×F : f 7→ f × id.
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Example 3. The bundle of rth order k-velocities is a natural bundle. On objects, T rk : M 7→T rkM := Jr0 (Rk,M) and on morphisms, T rk : f 7→ (jr0σ 7→ jr0f ◦ jr0σ). Dually, one can consider
the natural bundle T r∗k of rth order k-covelocities. On objects, T r∗k : M 7→ T r∗k M := Jr(M,Rk)0,
and on morphisms, T r∗k : f 7→ (jrσ 7→ jrσ ◦ jr0f−1), where f−1 is constructed locally.
Example 4. The frame bundles of natural vector bundles are, again, natural vector bundles.
Let E be a natural vector bundle with fiber dimension k and Fr(E) its frame bundle. On objects,
Fr(E) : M 7→ Fr(EM) := qx∈MLin(Rk ∼→ ExM). On morphisms, Fr(E) : f 7→ (σ 7→ Ef ◦ σ).
The following proposition follows trivially from the definition of a natural bundle.
Proposition 2. Let f : M → N be a local diffeomorphism of m-dimensional manifolds and let
x ∈M . For any natural bundle F , Ff |FxM : FxM → Ff(x)N depends only on the germ of f at
x.
Proof. Suppose g : M → N is a local diffeomorphism that agrees with f on an open submanifold
U ⊂M containing x. Let i denote the inclusion of U into M so that
f ◦ i = g ◦ i
and thus by functoriality Ff ◦Fi = Fg ◦Fi. By locality, Fi is the inclusion of π−1M (U) into FM .
Hence, Ff and Fg agree on π−1M (U) and, in particular, π−1
M (x).
A natural bundle has order r if its action on diffeomorphisms factors through r-jets. Formally,
Definition 7. A natural bundle F : Manm → Fib has order r, 0 ≤ r < ∞, if for every pair
of local diffeomorphisms f, g : M → N and every point x ∈ M , the equality jrxf = jrxg implies
Ff |FxM = Fg|FxM and if r is the minimal such integer.
Example 5. The tangent bundle has order 1. In coordinates, the pushforward map is given by
the Jacobian matrix, which depends exclusively on first-order derivatives. Similarly, the cotan-
gent and tensor bundles also have order 1.
Example 6. The trivial bundle ×F defined in Example 2 has order 0. Indeed, the behavior of
the map f × id at a point depends only on the value of f at that point.
Since sums, products, and tensor products are functorial constructions, we can form sums, prod-
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ucts, and tensor products of arbitrary natural bundles. These will, again, be natural bundles.
Furthermore, the order of a sum, product, or tensor product of two natural bundles of orders r1
and r2 will be the maximum of these two integers.
A foundational result in the theory of natural bundles is the finiteness theorem, proved by Palais
and Terng in [1].
Finiteness Theorem for Natural Bundles. If F is a natural bundle over m-dimensional
manifolds of fiber dimension k (i.e. dimF0Rm = k), then F has finite order less than 2k+1.
Note that this theorem is a significant strengthening of Proposition 2.
Another foundational result, which we state below without proof, is the regularity theorem for
natural bundles, originally proved by Epstein and Thurston in [3] and later generalized to natural
bundles of infinite-dimensional fiber by Mikulski in [9] .
Regularity Theorem for Natural Bundles. Natural bundles are regular in the following
sense: if F is a natural bundle and f : P ×M → N is a smoothly parameterized system of local
diffeomorphisms fp : M → N , then F f : P ×FM → FN , defined by (F f)p = Ffp, is a smoothly
parameterized system of local bundle isomorphisms.
5 Natural Operators
A natural operator η : F G between natural bundles F and G over m-dimensional manifolds
is a rule transforming smooth sections of F into smooth sections of G, in a way that respects the
functorial actions of F and G on diffeomorphisms. Formally,
Definition 8. Let F and G be natural bundles over m-dimensional manifolds. A natural op-
erator η : F G is a family of smooth maps ηM : C∞(FM) → C∞(GM), M ∈ Manm
satisfying
1. (Naturality) For every section s ∈ C∞(FM) and diffeomorphism f : M → N we have
ηN (Ff ◦ s ◦ f−1) = Gf ◦ ηM (s) ◦ f−1
2. (Locality) For every section s ∈ C∞(FM) and open submanifold U ⊂M we have
ηU (s|U ) = ηM (s)|U
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3. (Regularity) If ψ : P ×M → FM is a smoothly parameterized system of smooth sections
of F , the map ψ : P ×M → GM , defined by ψp = ηM (ψp), is a smoothly parameterized
system of smooth sections of G.
If F and G are, in particular, vector bundles, C∞(FM) and C∞(GM) are vector spaces. If, in
this case, the maps ηM : C∞(FM)→ C∞(GM) are linear (resp., polynomial), we say that η is
a linear (resp., polynomial) natural operator.
Put differently, a natural operator is a local differential operator, whose local coordinate descrip-
tion is invariant under arbitrary changes of coordinates.
Definition 9. Suppose F1, ..., Fk, G are natural vector bundles over m-dimensional manifolds.
By a multilinear natural operator η : F1 × F2 × ... × Fkmultilin G we mean a natural operator
η : F1 ×F2 × ...×Fk G for which the maps ηM : C∞(F1M)×C∞(F2M)× ...×C∞(FkM)→C∞(GM) are multilinear over R.
Remark 2. It is important to distinguish between multilinear natural operators F1 ×F2 × ...×Fk
multilin G, as defined above, and linear natural operators F1 ⊗ F2 ⊗ ... ⊗ Fk G. It is
well known that for any manifold M , C∞(F1M ⊗ F2M ⊗ ... ⊗ FkM) is canonically isomorphic
to C∞(F1M)⊗C∞(M) C∞(F2M)⊗C∞(M) ...⊗C∞(M) C
∞(FkM). In other words, linear natural
operators F1 ⊗F2 ⊗ ...⊗Fk G are multilinear over functions. Such operators correspond pre-
cisely to the punctual (order 0) multilinear operators F1×F2×...×Fkmultilin G. See Example 12.
We present a few simple examples of natural operators below.
Example 7. The exterior derivative d : ∧kT ∗ ∧k+1T ∗ is a linear natural operator transform-
ing k-forms into k+ 1-forms for every nonnegative integer k. Indeed, the exterior derivative has
a coordinate-free description given by:
dω(v0, v1, ..., vk) =∑i
(−1)iviω(v0, ..., vi, ..., vk) +∑i<j
(−1)i+jω([vi, vj ], v0, ..., vi, ..., vj , ...vk)
In fact, we will soon see that the exterior derivative is the only natural operator (up to scalar
multiplication) between k-forms and k + 1-forms.
Example 8. The wedge product ∧ : ∧aT ∗ × ∧bT ∗ ∧a+bT ∗ is a bilinear natural operator. It
has a coordinate-free description is given by:
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ω1 ∧ ω2(v1, v2, ..., va+b) =∑
(a,b)−shuffles
ω1(vσ(1), vσ(2), ..., vσ(a))ω2(vσ(a+1), vσ(a+2), ..., vσ(b))
The wedge product of alternating multivector fields ∧ : ∧aT × ∧bT ∧a+bT also defines a
bilinear natural operator with an analogous coordinate-free description.
Example 9. The Lie bracket [ , ] : T×T T is a bilinear natural operator. Indeed, if we regard
vector fields on M as derivations of the algebra C∞(M), the Lie bracket admits the following
coordinate-free description:
[X,Y ] = X ◦ Y − Y ◦X
In fact, for any pair of integers a, b ≥ 0, there is a bilinear natural operator [ , ] : ∧aT × ∧bT ∧a+b−1T called the Schouten-Nijenhuis bracket, which generalizes the Lie bracket (if a=b=0,
the bracket is identically zero). The Schouten-Nijenhuis bracket also admits a coordinate-free
expression. The Schouten-Nijenhuis bracket will receive quite a bit of attention in part 2.
Example 10. For integers a, b ≥ 0, the contraction i : ∧aT ∗ × ∧bT ∧a−bT ∗ is a bilinear
natural operator (where ∧−nT ∗ is shorthand for ∧nT ). Its coordinate-free description is given
by:
i(ω,X)(v1, ..., va−b) = ω(X ∧ v1 ∧ v2 ∧ ... ∧ va−b)
Definition 10. A natural operator η : F G has order r, 0 ≤ r <∞, if for every manifold M ,
point x ∈M , and section s ∈ C∞(FM), the value of ηM (s) at x depends only on jrx(s), and if r
is the minimal such integer.
Example 11. The exterior derivative d : ∧kT ∗ ∧k+1T ∗ has order 1. In coordinates,
d(fdx1 ∧ dx2 ∧ ... ∧ dxk) =∑i
∂f
∂xidx1 ∧ dx2 ∧ ... ∧ dxk ∧ dxi
which depends exclusively on first-order partials.
Example 12. The wedge product ∧ : ∧aT ∗×∧bT ∗ ∧a+bT ∗ has order 0. The value of ω1∧ω2
at a point depends entirely on the values of ω1 and ω2 at that point. Similarly for the wedge
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product of alternating multivector fields. Natural operators of order 0 are punctual.
Example 13. The Schouten-Nijenhuis bracket [ , ] : ∧aT × ∧bT ∧a+b−1T has order 1. In
the Lie bracket case, this can be seen in local coordinates:
[Xi∂i, Yi∂i] =
(Xj∂jY
i − Y j∂jXi)∂i
The general case can then be reduced to the Lie bracket and the wedge product via the following
formula:
[X1∧X2∧ ...∧Xa, Y1∧Y2∧ ...∧Yb] =∑i,j
(−1)i+j [Xi, Yj ]X1∧ ...∧Xi∧ ...∧Xa∧Y1∧ ...∧ Yj∧ ...∧Yb
Example 14. The contraction i : ∧aT ∗×∧bT ∧a−bT ∗ is of course order 0. It is given locally
by the evaluation of a form at a multivector field.
For every fiber bundle π : E → M and natural number r, there is a fiber bundle Jr(π) → M
consisting of r-jets of smooth sections of π.
Definition 11. Let π : E → M be a fiber bundle and r a natural number. The rth order jet
bundle over π is the set {jrxs | x ∈ M, s ∈ C∞(E)} and is denoted by Jr(π) or, infrequently,
Jr(E) when the projection map is implied. The source projection jrxs 7→ x makes Jr(π) into a
fiber bundle over M .
Note that for every natural bundle F : Manm → Fib, the assignment M 7→ Jr(FM) defines a
covariant functor JrF : Manm → Fib. Indeed, for every local diffeomorphism f : M → N , there
is a smooth bundle map JrF (f) : Jr(FM)→ Jr(FN) given by JrF (f)(jrxs) = jrf(x)(Ff◦s◦f−1),
which respects compositions. It is easy to see that JrF satisfies both criteria in Definition 6 .
Thus, JrF is a natural bundle, which we call the rth order jet prolongation of F .
Proposition 3. Suppose F is a natural bundle of order k. Then JrF is a natural bundle of
order k + r.
Proof. For any local diffeomorphism f : M → N , we have
JrF (f)(jrxs) = jrf(x)(Ff ◦ s ◦ f−1)
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Since jrs(x)(Fjkxf) = Fjr+kx (f), it follows that JrF (f)(jrxs) depends only on the (k + r)-jet of f
at x.
Suppose F and G are natural bundles of order k and l, respectively. Every r-order natural
operator η : F G induces a natural transformation η : JrF → G defined by the formula
ηM (jrxs) = ηM (s)(x)
The smoothness of the components follows from the regularity of η. The naturality of η follows
from the naturality of η.
Proposition 4. The correspondence η 7→ η defines a bijection from natural r-order operators
F G onto natural transformations JrF → G.
Proof. For every natural transformation ξ : JrF → G define an operator ξ : F G by the
formula
ξM (s)(x) = ξM (jrxs)
Since the components of ξ are smooth (locally invertible) bundle maps, the maps x 7→ ξM (jrxs)
are smooth sections of GM . Since all categories in question are local, the induced operator is
local. Moreover, for every section s ∈ C∞(FM) and diffeomorphism f : M → N , we have, by
the naturality of ξ,
ξN (Ff ◦ s ◦ f−1)(f(x)) = ξN (jrf(x)(Ff ◦ s ◦ f−1))
= ξN (JrF (f)(jrxs))
= (ξN ◦ JrFf)(jrxs)
= (Gf ◦ ξM )(jrxs)
= Gf(ξM (s)(x))
= (Gf ◦ ξMs ◦ f−1)(f(x))
Hence, ξ is natural. To verify the regularity of ξ, consider a smoothly parameterized family of
smooth sections of F , ψ : P ×M → FM . ξ induces a map ψ : P ×M → GM given by
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ψ(p, x) = ξM (jrxψp)
Since ξ is a natural transformation, ξM is smooth. Hence, it suffices to show that the assignment
(p, x) 7→ jrxψp is smooth. In charts, jrxψp is just an rth order Taylor polynomial. In other words,
the map (p, x) 7→ jrxψp is, locally, a polynomial over derivatives of the smooth function ψ. Thus,
ψ is manifestly smooth.
We should, at this point, pause to remark that there is no general analogue of Palais-Terng
finiteness for natural operators. However, in many applications, the natural operators in question
are presumed to be linear. In these cases, finiteness often follows from the Invariant Tensor
Theorem (see section 7). In other situations, we will appeal to partial results, such as theorem
3 which we state without proof in section 6.
6 Invariant-Theoretic Reduction
The methods of invariant-theoretic reduction allow us to regard natural bundles over manifolds
as modules over jet groups and natural operators as module homomorphisms, thus reducing the
geometric problem of classifying natural bundles and operators to an essentially algebraic one.
We begin with a few preliminary observations.
Definition 12. The standard fiber of a natural bundle F is the fiber F0Rm over the origin.
Proposition 5. Let F be a natural bundle with standard fiber S. The fiber bundle FRm is
canonically isomorphic to the trivial bundle Rm × S.
Proof. For every x ∈ Rm, let tx : Rm → Rm denote the parallel shift tx(a) = x+ a. There is an
obvious bundle map ϕ : Rm × S → FRm given by
ϕ(x, s) = Ftx(s)
which is smooth by the regularity of F . In fact, ϕ is an isomorphism. Its inverse is given by
ϕ−1(v) = (π(v), F t−π(v)(v))
16
Replacing the parallel shifts in the proof above with arbitrary chart maps, we get
Proposition 6. For every m-dimensional manifold M , the value FM is a locally trivial fiber
bundle with standard fiber S.
Proof. Let a ∈M and let φ : U → Rm be a chart map with φ(a) = 0. By the locality condition,
FU is π−1M U . Hence, the value Fφ is a diffeomorphism from π−1
M U onto FRm. Left composing
with the isomorphism in Proposition (5), we obtain a bundle isomorphism ϕ−1 ◦ Fφ : π−1M U →
Rm × S.
If F is a natural bundle of order less than or equal to r, there is a canonical (left) action of the
jet group Grm on the standard fiber S given by the formula
jr0(f)s = Ff(s)
for every jr0(f) ∈ Grm and s ∈ S. We call this the induced action of Grm. By the regularity of F ,
the induced action is smooth.
Theorem 1. Restricting to the standard fiber defines a bijective correspondence between (equiv-
alence classes of) natural bundles of order less than or equal to r and smooth left actions of Grm
on smooth manifolds.
Proof. We have already seen that for every natural bundle F of order not exceeding r, there is a
smooth left action of Grm on its standard fiber, S. We must exhibit a map in the other direction.
Let S be a manifold and ϕ : Grm × S → S a smooth left action. We begin by considering the
rth order frame bundle Grm(M) over an m-dimensional manifold M . Grm(M) is the set of r-jets
jrx(f) ⊂ Jrx(M,Rm)0 such that dfx maps TxM isomorphically onto Rm = T0Rm. As such, it is
an open submanifold of the bundle of rth order m-covelocities T r∗m (M) (see Example (3)). The
projection jrx(f) 7→ x turns Grm(M) into a fiber bundle over M . Moreover, there is a smooth
right action of Grm on Grm(M) given by
jrx(f)jr0(ϕ) = jrx(ϕ−1 ◦ f)
Clearly, this action is free and transitive on the fibers of Grm(M). Hence, Grm(M) is a principal
Grm-bundle over M . We then then take the product Grm(M)× S and let Grm act on the left via
jr0(ϕ)(jrx(f), s) = (jrx(f)jr0(ϕ−1), jr0(ϕ)s)
17
denoting the orbit space of this action by S(M). We will quickly verify that S(M) is a natural
bundle over m-dimensional manifolds.
For any local diffeomorphism φ : M → N , there is an induced map Grmφ : Grm(M) → Grm(N),
defined by jrx(f) 7→ jrx(f◦φ−1), which is Grm-invariant by construction. Therefore, so is Grm×idS :
Grm(M) × S → Grm(N) × S, which induces a smooth bundle map Sφ : S(M) → S(N). The
naturality, locality, and regularity of S(M) follow trivially from the naturality, locality, and
regularity of Grm(M). Moreover, it is clear that the order of S(M) is less than or equal to r.
The standard fiber of S(M) is the set of equivalence classes {[id, s] | s ∈ S}, which we identify
canonically with S, and the canonical action of Grm on S0(Rm) is indeed jr0(ϕ)[id, s] = [id, jr0(ϕ)s].
On the other hand, if we start with a natural bundle F with standard fiber S, the functors
F : Manm → Fib and S : Manm → Fib are naturally equivalent.
The equivalence is defined as follows. For every manifold M and point x ∈ M , we pick a local
diffeomorphism ψ : Rm →M with ψ(0) = x. Then, for every y ∈ FxM , we put
ηM (y) = S0(ψ)[id, F0(ψ)−1(y)]
Note that these mappings are independent of ψ. Indeed, if φ is a another local diffeomorphism
φ : Rm →M with φ(0) = x, we have, for every s ∈ S,
[id, F0(φ−1 ◦ ψ)(s))] = S0(φ−1 ◦ ψ)[id, s]
and so by the functoriality of F and S,
S0(φ)[id, F0(φ−1)(s)] = S0(ψ)[id, F0(ψ−1)(s)]
Now, let g : M → N be a local diffeomorphism. For every x ∈ M , y ∈ FxM , and choice of
local diffeomorphism ψ : Rm → M with ψ(0) = x, the composition g ◦ ψ : Rm → N is a local
diffeomorphism with (g ◦ ψ)(0) = g(x) and
ηN (Fxg(y)) = S0(g ◦ ψ)[id, F0(g ◦ ψ)−1(Fxg(y))]
= S0(g)(S0(ψ)[id, F0(ψ)−1(y)])
= S0(g)(ηM (y))
Hence, η defines a natural isomorphism η : F∼→ S, which completes the proof.
18
If F is, in particular, a natural vector bundle (i.e. F takes values in the subcategory of vector
bundles over m-dimensional manifolds with locally invertible, fiber-wise linear bundle maps),
then the standard fiber of F is a vector space and the action of Grm is linear. Conversely, every
Grm-module S gives rise to a natural vector bundle with standard fiber S and order not exceeding
r through the construction described above. Therefore,
Corollary 1. Restricting to the standard fiber defines a bijective correspondence between (equiv-
alence classes of) natural vector bundles of order less than or equal to r and Grm-modules.
Example 15. The tangent bundle T is a natural vector bundle of order 1 and standard fiber
Rm. G1m = GLm(R) acts on Rm canonically, i.e.
g • v = gv
The cotangent bundle T ∗ is a natural vector bundle of order 1 and standard fiber Rm∗. GLm(R)
acts on Rm∗ canonically, i.e.
g • v = vg−1
For every tensor bundle ⊗pT ⊗⊗qT ∗, the corresponding GLm(R)-module is the tensor product
⊗pRm ⊗⊗qRm∗, with the canonical linear action
g • (v1 ⊗ ...⊗ vp ⊗ w1 ⊗ ...⊗ wq) = gv1 ⊗ ...⊗ gvp ⊗ w1g−1 ⊗ ...⊗ wqg−1
Example 16. Suppose F is a natural bundle of order k and standard fiber S. Then the rth
order jet bundle Jr(F ) is a natural bundle of order k+r with standard fiber T rm(S) := Jr0 (Rm, S)
(see Example (3)). The action of the jet group Gr+km on T rm(S) is inherited directly from S, i.e.
g • jr0(f) = jr0(g • f)
In the special case where F is a vector bundle with, say, standard fiber V , T rm(V ) is canonically
identified as a vector space (and, indeed, a GLm(R)-module) with the direct sum
V ⊕ Rm∗ ⊗ V ⊕ S2Rm∗ ⊗ V ⊕ ...⊕ SrRm∗ ⊗ V
19
To see this, select a basis {v1, ...vn} for V . Then, the set {∂|α|fi
∂xα }i,α forms a basis for T rm(V ).
The subspace Sj(Rm) ⊗ V coincides with the linear span of basis elements ∂|α|fi
∂xα with |α| = j
(all products are symmetric since mixed partials commute). This identification is independent
of our particular choice of basis.
Proposition 4 says that natural operators are essentially natural transformations between natural
bundles. Theorem (1) then says that natural bundles can be regarded as smooth Grm-spaces. It
is not surprising, therefore, that natural operators can be regarded as smooth, equivariant maps
between smooth Grm-spaces (or, in the vector bundle case, Grm-module homomorphisms).
Theorem 2. Let F andG be natural bundles of order k and l, respectively, and let r = max (k, l).
Then restriction defines a bijective correspondence between natural transformations F → G and
smooth, Grm-equivariant maps between their standard fibers.
Proof. Let S and Q be the standard fibers of F and G, respectively. Every natural transformation
η : F → G induces a smooth map ηRm |S : S → Q, which is Grm-equivariant, by the naturality of
η.
Conversely, given any Grm-equivariant map f : S → Q, we can define a natural transformation
η : F → G by pushing forward and backward along locally invertible chart maps. Specifically, for
every manifold M , point x ∈ M , and locally invertible chart map ψ : M → Rm with ψ(x) = 0,
we put
ηM |FxM := G(ψ−1)|Q ◦ f ◦ F (ψ)|FxM (2)
which is smooth, since f is smooth, and which is independent of ψ by the funtoriality of F and G
and the equivariance of f . The components ηM , formed by assembling these maps, are smooth,
since the chart maps {Fψ} and {Gψ} form smooth bundle atlases for FM and GM , respectively.
Moreover, for any local diffeomorphism g : M → N , the diagram
FM FN
GM GN
Fg
ηM ηN
Gg
is easily seen to commute. Indeed, for any y ∈ FxM , the composition ψ ◦ g−1 : N → Rm is a
locally invertible chart map with (ψ ◦ g−1)g(x) = 0 and
20
ηN (Fg(y)) = G(g ◦ ψ−1)(f(F (ψ ◦ g−1)(Fg(y))))
= Gg(G(ψ−1)(f(F (ψ)(y)))
= Gg(ηM (y))
Thus, η : F → G is a natural transformation. Clearly, the restriction of η to S is simply f , the
equivariant map we started with. On the other hand, every natural transformation η : F → G
which restricts to f on S satisfies equation (2). Thus, if we start with a natural transformation
η : F → G, the natural transformation constructed via (2) from the restriction of η to S is
precisely η. Hence, the correspondence is a bijection, and we are done.
Combining this with the correspondence established in Proposition (4), we get
Corollary 2. Let F be a k-order natural bundle with standard fiber S, and let G be an l-
order natural bundle with standard fiber Q. Then restriction defines a bijective correspondence
between natural r-order operators F G and smooth Gpm-equivariant maps T rm(S)→ Q, where
p = max (k + r, l).
In the vector bundle case, we can apply the identification described in Example (16)
Corollary 3. Let F be a k-order natural vector bundle with standard fiber V , and let G be
an l-order natural vector bundle with standard fiber W . Then restriction defines a bijective
correspondence between natural r-order operators F G and Gpm-equivariant maps
V ⊕ Rm∗ ⊗ V ⊕ ...⊕ SrRm∗ ⊗ V →W
where p = max (k + r, l).
In summary, we have proved:
Natural r-order Operators
F G
Natural Transformations
JrF → G
Smooth Gpm-equivariant maps
T rm(S)→ Q
∼ ∼
Remark 3. Suppose F , G, and H are natural vector bundles (of order k, l, and n and with
standard fibers S, Q, and R) and F × G H is an r-order natural bilinear operator. Let p =
21
max (r + k, r + l, n). The fibered product JrF×JrG has order max (r + k, r + l) and the induced
natural transformation JrF × JrG → H restricts to a bilinear Gpm-module homomorphism
T rm(S) × T rm(Q) → R. Conversely, a bilinear Gpm-module homomorphism T rm(S) × T rm(Q) → R
gives rise to a bilinear natural operator F × G H through the procedure described in the
proof of Theorem (2). In other words, there is a bijective correspondence between natural r-
order bilinear operators F×G H and linear Gpm-module homomorphisms T rm(S)⊗T rm(Q)→ R.
Similarly, natural multilinear operators F1 × F2 × ... × Fn G correspond bijectively to linear
module homomorphisms T rmS1 ⊗ T rmS2 ⊗ ...⊗ T rmSn → Q.
Let us reexamine some of our original examples of natural operators through an invariant-
theoretic lens.
Example 17. The first-order vector bundle ∧kT ∗ corresponds to the vector space ∧kRm∗, with
the canonical action of GLm(R). Thus, the jet bundle J1(∧kT ∗) corresponds to the G2m-module
T 1m(∧kRm∗) = ∧kRm∗ ⊕ Rm∗ ⊗ ∧kRm∗
The exterior derivative d : ∧kT ∗ ∧k+1T ∗ is then the linear G2m-module homomorphism
d : ∧kRm∗ ⊕ Rm∗ ⊗ ∧kRm∗ → ∧k+1Rm∗
dxi1 ∧ dxi2 ∧ ... ∧ dxik + dxj0 ⊗ dxj1 ∧ dxj2 ∧ ... ∧ dxjk 7→ dxj0 ∧ dxj1 ∧ ... ∧ dxjk
Example 18. The wedge product of differential forms ∧ : ∧aT ∗ ×∧bT ∗ ∧a+bT ∗ corresponds
to the GLm(R)-module homomorphism
∧ : ∧aRm∗ ⊗ ∧bRm∗ → ∧a+bRm∗
dxi1 ∧ dxi2 ∧ ... ∧ dxia ⊗ dxj1 ∧ dxj2 ∧ ... ∧ dxjb 7→ dxi1 ∧ ...dxia ∧ dxj1 ∧ ... ∧ dxjb
Similarly for the wedge product of alternating multivector fields.
Example 19. The Schouten-Nijenhuis bracket [ , ] : ∧aT ×∧bT ∧a+b−1T corresponds to the
G2m-module homomorphism
[ , ] : (∧aRm ⊕ Rm∗ ⊗ ∧aRm)⊗ (∧bRm ⊕ Rm∗ ⊗ ∧bRm)→ ∧a+b−1Rm
22
given by
[ , ] : (∂i1 ∧ ... ∧ ∂ia + dxj0 ⊗ ∂j1 ∧ ... ∧ ∂ja)⊗ (∂k1 ∧ ... ∧ ∂kb + dxl0 ⊗ ∂l1 ∧ ... ∧ ∂lb)
7→a∑l=1
(−1)ldxl0(∂il)∂i1 ∧ ... ∧ ∂il ∧ ... ∧ ∂ia ∧ ∂l1 ∧ ... ∧ ∂lb
−(−1)(a−1)(b−1)b∑l=1
(−1)ldxj0(∂kl)∂j1 ∧ ... ∧ ∂ja ∧ ∂k1 ∧ ... ∧ ∂kl ∧ ... ∧ ∂kb−1
Example 20. The contraction i : ∧aT ∗ × ∧bT ∗ ∧a−bT ∗ (where ∧−nT ∗ = ∧nT ) corresponds
to the GLm(R)-homomorphism
i : ∧aRm∗ ⊗ ∧bRm → ∧a−bRm∗
dxi1 ∧ ... ∧ dxia ⊗ ∂j1 ∧ ... ∧ ∂jb 7→∑σ∈Sa
sgn(σ)dxσ(i1)(∂j1)...dxσ(ib)(∂jb)dxσ(ib+1) ∧ ... ∧ dxσ(ia)
as always, reinterpreting in the obvious way if a < b.
The following is a partial finiteness theorem for natural operators.
Theorem 3. Let F and G be natural bundles corresponding, via the identification described in
Theorem 1, to GLm(R)-submodules of the finite sums
⊕i
(⊗ai
Rm ⊗⊗bi
Rm∗)
and
⊕j
⊗a′j
Rm ⊗⊗b′j
Rm∗
respectively. Then every natural operator η : F G has finite order.
Proof. See Proposition 23.5 in [5].
23
7 Classical Results
In this section, we state and prove a variety of classical results. Chief among these are the Ho-
mogenous Function Theorem and Weyl’s celebrated Invariant Tensor Theorem. These two results
are the most powerful tools in our arsenal for describing and classifying natural differential oper-
ators, and a large class of operators can be exhaustively classified using these two theorems alone.
Homogenous Function Theorem. Let V1, V2, ..., Vn,W be finite dimensional vector spaces
and f : V1 × ...× Vn →W a smooth function. Let a1, ..., an, b be real numbers with ai > 0 such
that
kbf(v1, ..., vn) = f(ka1v1, ..., kanvn) (3)
for every n-tuple (v1, ..., vn) and k > 0. Then f is a sum of polynomials of degree di in vi such
that
a1|d1|+ ...+ an|dn| = b
If no such non-negative integers d1, ..., dn exist, then f is identically 0.
Proof. First observe that if f satisfies (3) with b < 0, then f is identically 0. Indeed, if
f(x1, ..., xn) 6= 0 for some x1, ..., xn, then taking limits k → 0+, we get
limk→0+
kbf(x1, ..., xn) = limk→0+
f(ka1x1, ..., kanxn) = f(0, ..., 0)
But since b < 0, the limit on the left is improper.
Suppose now b ≥ 0. Assume, without loss of generality, that a1 = min(a1, ..., an), and set p
equal to the integer part of ba1
. Select a basis xji on each Vi. Differentiating (3) with respect to
xji , we get
kbfxji (v1, ..., vn) = kaifxji (ka1v1, ..., k
anvn)
Thus, the partial derivative fxji satisfies the homogeneity condition
kb−aifxji (v1, ..., vn) = fxji (ka1v1, ..., k
anvn)
24
Iterating this argument, we find that every partial derivative of order p + 1 or greater satisfies
(3) with b < 0. Hence, by the argument above, all such partials are identically 0. It follows from
Taylor’s theorem that f is a polynomial of maximal order r. Suppose vt11 ...vtnn is a monomial of
degree |ti| in vi. Then,
(ka1v1)t1 ...(kanvn)tn = ka1|t1|+...+an|tn|vt11 ...vtnn
Since f satisfies (3) for all k > 0, it follows that a1|t1|+ ...+ an|tn| = b.
The Homogenous Function Theorem can often be used to demonstrate the linearity of certain
types of operators.
Example 21. Suppose η : ∧kT ∗ ∧k+1T ∗ is a natural operator and k > 1. By Theorem 3, η
is of finite order r. Thus, η corresponds to a Gr+1m -module homomorphism
f :
r⊕i=0
∧kRm∗ ⊗ Si(Rm∗)→ ∧k+1Rm∗
The jet group Gr+1m contains the subgroup GLm(R) = G1
m. The linear homotheties {t(Idm)|t ∈R∗}, in turn, form a subgroup of GLm(R). Naturality with respect to linear homotheties defines
a homogeneity condition
tk+1f(v0, v1, ..., vr) = f(tkv0, tk+1v1, ..., t
k+rvr) ∀t 6= 0
Therefore, by the Homogenous Function Theorem, f is a sum of polynomials of degree di in vi
such that
k|d0|+ (k + 1)|d1|+ ...+ (k + r)|dr| = k + 1
For k > 1, the only solution is given by |d1| = 1 and |di| = 0 ∀i 6= 1. Hence f is a linear function
in the second component ∧kRm∗ ⊗ Rm∗ and η is a linear first-order operator.
Every polynomial map f : V →W between finite-dimensional vector spaces decomposes uniquely
into homogenous components
f = f0 + f1 + ...+ fn
25
Proposition 7. If V and W are finite-dimensional G-modules and the polynomial map f : V →W is G-equivariant, then each homogenous component is also G-equivariant.
Proof. For every g ∈ G, we have, by the linearity of the G-action
gf0 + ...+ gfn = f0g + ...+ fng
In a basis, multiplying by g corresponds to a linear change of coordinates. Thus, for each fi,
both gfi and fig are homogenous of degree i. Hence, gfi = fig for every g ∈ G.
Similarly, every polynomial map f : V1 × ... × Vp → W decomposes uniquely into multi-
homogenous components
f =∑
(d1,...,dp)
f(d1,...,dp)
where the multi-homogenous degree of a monomial xα11 ...xαnn in f is the n-tuple (|α1|, ..., |αn|).
Proposition 8. If V1, ..., Vp,W are finite-dimensional G-modules and the polynomial map f :
V1×...×Vp →W is G-equivariant, then each multi-homogenous component is also G-equivariant.
Proof. For every g ∈ G, we have, by the linearity of the G-action
∑(d1,...,dp)
gf(d1,...,dp) = gf = gf =∑
(d1,...,dp)
f(d1,...,dp)g
In a basis, multiplying by g corresponds to a linear change of coordinates. Thus, for each
f(d1,...,dp), both gf(d1,...,dp) and f(d1,...,dp)g are multi-homogenous of multi-homogenous degree
(d1, ..., dp). Hence, gf(d1,...,dp) = f(d1,...,dp)g for every (d1, ..., dp).
Let V be a finite-dimensional G-module. The induced action of G on the dual space V ∗ is given
by
(gw)(v) = w(g−1v) ∀w ∈ V ∗, v ∈ V, g ∈ G
The tensor product V ⊗W of G-modules is a G-module, where
g(v ⊗ w) = gv ⊗ gw
26
In particular, every tensor product ⊗pV ⊗⊗qV ∗ is a canonical GL(V )-module.
Definition 13. A tensor x ∈ ⊗pV ⊗ ⊗qV ∗ is invariant if it is a fixed point of the canonical
GL(V ) action.
There is a natural isomorphism
⊗pV ⊗⊗qV ∗ ∼= Lin (⊗qV,⊗pV )
given by
v1 ⊗ ...⊗ vp ⊗ w1 ⊗ ...⊗ vq 7→ (x1 ⊗ ...⊗ xq 7→ w1(x1)...wq(xq)v1 ⊗ ...⊗ vp)
If, in particular, p = q, there is an obvious collection of invariant tensors given by permuations of
indices. Explicity, for every σ ∈ Sp, consider the tensor Iσ identified with the linear map
Iσ : x1 ⊗ ...⊗ xp 7→ xσ(1) ⊗ ...⊗ xσ(p)
Clearly, Iσ is GL(V )-invariant. Invariant tensors defined in this way are called elementary in-
variant tensors.
Invariant Tensor Theorem. Let I denote the subspace of invariant tensors in ⊗pV ⊗⊗qV ∗.Then
1. If p 6= q, I = 0.
2. If p = q, I is generated by the elementary invariant tensors {Iσ}σ∈Sp .
3. If p = q and dimV ≥ p, the elementary invariant tensors {Iσ}σ∈Sp are linearly independent
and, thus, form a basis for I.
Proof. We closely follow the proof of proposition 24.4 in [5].
For 1, consider the linear homothety k(Id) for k 6= 0. For any x ∈ I, invariance with respect to
k(Id) implies kp−qx = x. Thus, if p 6= q, we have no choice but x = 0.
The only interesting case occurs when p = q. Let X = ei1...ipj1...jp
be an element of I and put
m = dimV . Invariance means
ai1...ipk1...kp
ek1...kpl1...lp
= ei1...ipj1...jp
aj1...jpl1...lp
27
for every matrix aij ∈ GLm(R). Equivalently,
aj1...jpk1...kp
δ(i1, j1)...δ(ip, jp)ek1...kpl1...lp
= ei1...ipj1...jp
δ(k1, l1)...δ(kp, lp)aj1...jpk1...kp
Since this holds for every aij ∈ GLm(R), we can compare coefficients in each monomial in aij to
obtain
∑σ∈Sp
δ(i1, jσ(1))...δ(ip, jσ(p))ekσ(1)...kσ(p)l1...lp
=∑σ∈Sp
δ(kσ(1), l1)...δ(kσ(p), lp)ei1...ipjσ(1)jσ(p)
(4)
If m ≥ p, put cσ = e1...pσ(1)...σ(p) and i1 = 1 = j1, ..., ip = p = jp so that the only non-zero term on
the left-hand side of equation (4) is given by σ = 1. Hence
ek1...kpl1...lp
=∑σ∈Sp
cσδ(kσ(1), l1)...δ(kσ(p), lp) (5)
which establishes 1 in the special case m ≥ p. Furthermore, since the coefficients cσ in (4) are
uniquely determined by ei1...ipj1...jp
, we have also proved 2.
If, on the other hand, m < p, a subtler argument is needed. In this case, the coefficients cσ in
(5) are not uniquely determined, since for m < p, the system of m2p equations in p! variables dσ
∑σ∈Sp
δ(i1, jσ(1))...δ(ip, jσ(p))dσ = 0 (6)
has nontrivial solutions. For example, the alternation
∑σ∈Sp
sgn(σ)δ(i1, jσ(1))...δ(ip, jσ(p))
is 0, by the pigeonhole principle. Select a basis dασ , α = 1, ..., t for the space of solutions to (6)
and consider the equations in dσ
∑σ∈Sp
dασdσ = 0 α = 1, ..., t (7)
We claim the system (6) and (7) has rank p! (i.e. full column rank). To this end, it suffices to
show that the only solution to equations (6) and (7) is trivial. If dσ satisfy (6), there are kα such
that, for every σ ∈ Sp,
28
dσ =
t∑α=1
kαdασ (8)
If dσ also satisfy (7), we have
t∑α=1
kα∑σ∈Sp
(dασdσ) = 0 (9)
Rewriting the equation above and applying equation (8), we obtain
0 =
t∑α=1
kα∑σ∈Sp
(dασdσ) =∑σ∈Sp
dσ
t∑α=1
kαdασ =
∑σ∈Sp
d2σ (10)
It follows that all dσ are 0. Thus, we have the following lemma:
Lemma 1. Suppose p! tensors Xσ ∈ ⊗pRm ⊗⊗qRm∗ satisfy the equations
∑σ∈Sp
δ(i1, jσ(1))...δ(ip, jσ(p))Xσ =∑σ∈Sp
ci1...ipjσ(1)...jσ(p)
Iσ (11)
with real coefficients ci1...ipjσ(1)...jσ(p)
and
∑σ∈Sp
dασXσ = 0 α = 1, ..., t (12)
with dασ as above. Then every Xσ is a linear combination of {Iσ}.
Indeed, since the system (6) and (7) has full rank p! and the equations in (7) are linearly
independent, there is a subsystem (7’) of (7) such that the system (6) and (7’) is invertible. Let
(11’) denote the subsystem of (11) corresponding to (7’) so that the system (11’) and (12) is also
invertible. Applying Cramer’s rule, we see that each Xσ must be a linear combination of the
right-hand sides of equations (11) and (12), and hence, a linear combination of {Iσ}.
With this lemma in hand, our proof quickly falls into place. Let X be an element of I, and for
every σ ∈ Sp, let Xσ be the result of applying σ to X. Then, equation (4) translates to
∑σ∈Sp
δ(i1, jσ(1))...δ(ip, jσ(p))Xσ =∑σ∈Sp
δ(kσ(1), l1)...δ(kσ(p), lp)Iσ (13)
29
Since dασ are solutions to (6), the tensor
∑σ∈Sp
δ(i1, jσ(1))...δ(ip, jσ(p))dασ
is 0. Contracting with X we obtain
∑σ∈Sp
dασXσ = 0 (14)
Applying our lemma to equations (13) and (14), we conclude that each Xσ, and, therefore X in
particular, is a linear combination of {Iσ}.
We can apply the Homogenous Function Theorem and Invariant Tensor Theorem in tandem to
deduce all GL(V )-equivariant maps between symmetric and alternating powers of V .
Let Sym : ⊗pV → Sp(V ) and Alt : ⊗p → ∧p(V ) be the symmetrization and alternation maps,
respectively. Consider also the inclusions Sp(V ) ↪→ ⊗pV and ∧p(V ) ↪→ ⊗p(V ) which we denote
in both cases by i. Note that all four maps are linear and GL(V )-equivariant. Let f : Sp(V )→Sp(V ) be a smooth GL(V )-equivariant map and put f := i◦f ◦Sym. Since Sym : ⊗pV → Sp(V )
is a left inverse for i : Sp(V ) ↪→ ⊗pV , the following diagram commutes
Sp(V ) Sp(V )
⊗p(V ) ⊗p(V )
f
i
f
Sym
Since the maps Sym, i, and f are smooth and GL(V )-equivariant, so is f . Invariance with
respect to homotheties, as in example 21, yields an homogeneity condition
kp(f(x)) = f(kpx) ∀k 6= 0
which, by application of the Homogenous Function Theorem, implies that f is linear. Therefore,
via the obvious identification ⊗p(V )⊗⊗p(V ∗) ∼= Lin(⊗pV,⊗pV ), f corresponds to an invariant
tensor in the vector space ⊗p(V )⊗⊗p(V ∗). It follows from the Invariant Tensor Theorem that
f is a linear combination of elementary invariant tensors {Iσ}σ∈Sp . Take some Iσ and consider
the composition Sym ◦ Iσ ◦ i. We have, for every v1 ⊗ ...⊗ vp ∈ Sp(V ),
30
Sym ◦ Iσ ◦ i(v1 ⊗ ...⊗ vp) = Sym ◦ Iσ(v1 ⊗ ...⊗ vp)
= Sym(v1 ⊗ ...⊗ vp)
= v1 ⊗ ...⊗ vp
Hence f is a constant multiple of the identity. Reproducing this reasoning with different types
of smooth, equivariant maps of variously alternating and symmetric domain and codomain, we
obtain the following classification:
Proposition 9. Every smooth, GL(V )-equivariant map
1. Sp(V )→ Sp(V ) is a constant multiple of the identity
2. ∧p(V )→ ∧p(V ) is a constant multiple of the identity
3. Sp(V )→ ⊗p(V ) is a constant multiple of the inclusion
4. ∧p(V )→ ⊗p(V ) is a constant multiple of the inclusion
5. ⊗p(V )→ Sp(V ) is a constant multiple of the symmetrization
6. ⊗p(V )→ ∧p(V ) is a constant multiple of the alternation
7. Sp(V )→ ∧p(V ) is identically 0
8. ∧p(V )→ Sp(V ) is identically 0
The final two claims can be strengthened considerably.
Proposition 10. Let M be a GL(V )-submodule of ⊗qV such that the inclusion M ↪→ ⊗qVadmits a linear GL(V )-equivariant left inverse (standard examples include ∧q(V ), Sq(V ) and
∧a(V )⊗ Sq−a(V )). Then, for any integer p > 1, every smooth, GL(V )-equivariant map
1. Sp(V )⊗M → ∧r(V ) is identically 0
2. ∧p(V )⊗M → Sr(V ) is identically 0
Proof. Let f : Sp(V ) ⊗M → ∧r(V ) be a smooth, GL(V )-equivariant map. Acting by homoth-
eties, we obtain the homogeneity condition
krf(x) = f(kp+qx) ∀k 6= 0
31
By the Homogenous Function Theorem, f is identically 0 if (p+ q) - r. Otherwise, (p+ q)n = r
and f is a linear map
f : Sn (Sp(V )⊗M)→ ∧r(V )
Let φ : ⊗q(V )→M be a linear GL(V )-equivariant left inverse of the inclusion M ↪→ ⊗qV , which
is guaranteed by assumption, and consider the map
Φ := Sn(Sym⊗ φ) : ⊗r(V )→ Sn (Sp(V )⊗M)
Let f := i ◦ f ◦ Φ : ⊗r(V )→ ⊗r(V ) so that the following diagram commutes
Sn (Sp(V )⊗M) ∧r(V )
⊗r(V ) ⊗r(V )
f
i
f
Alt
As a composition of linear, GL(V )-equivariant maps, f is linear and GL(V )-equivariant. Hence,
by the Invariant Tensor Theorem, f is a linear combination of elementary invariant tensors. Take
one such tensor Iσ. Since p > 1 and n ≥ 1, for every v1 ⊗ ...⊗ vr ∈ Sn (Sp(V )⊗M), the tensor
Iσ(v1 ⊗ ... ⊗ vr) is symmetric in at least two components. Hence, Alt(Iσ(v1 ⊗ ... ⊗ vr) = 0, as
desired. The proof of 2 is analogous.
8 Natural Operators on Differential Forms
In this section, we apply invariant-theoretic reduction and the results of section 7 to classify all
natural operators of type
∧p1T ∗ × ...× ∧pnT ∗ ∧qT ∗
on m-dimensional manifolds. For the following lemma, let � denote the symmetric tensor prod-
uct.
Lemma 2. Let f :⊙n
i=1 ∧pi(V )⊗ Sqi(V )→ ∧r(V ) be a smooth, GL(V )-equivariant map with∑ni=1 pi+qi = r. Then f = 0 if any qi > 1. Otherwise, f is a constant multiple of the alternation.
32
Proof. Our proof is quite similar to proposition 10. Acting by homotheties, we obtain the
homogeneity condition
krf(x) = f(krx) ∀k 6= 0
Thus, by the Homogenous Function Theorem, f is necessarily linear.
The inclusion⊙n
i=1 ∧pi(V )⊗ Sqi(V ) ↪→ ⊗rV admits an equivariant left inverse
Φ :=
n⊙i=1
Alt⊗ Sym : ⊗r(V )→n⊙i=1
∧pi(V )⊗ Sqi(V )
Put f := i ◦ f ◦ Φ so that the following diagram commutes
⊙ni=1 ∧pi(V )⊗ Sqi(V ) ∧r(V )
⊗r(V ) ⊗r(V )
f
i
f
Alt
As a composition of linear, GL(V )-equivariant maps, f is linear and GL(V )-equivariant. Hence,
by the Invariant Tensor Theorem, f is a linear combination of elementary invariant tensors.
Take one such tensor Iσ. If qi > 1, then every v1 ⊗ ...⊗ vr ∈⊙n
i=1 ∧pi(V )⊗ Sqi(V ), and hence
Iσ(v1 ⊗ ...⊗ vr), is symmetric in at least two components. Hence, Alt(Iσ(v1 ⊗ ...⊗ vr)) = 0, as
desired. Otherwise,
Alt(Iσ(v1 ⊗ ...⊗ vr)) = Alt(vσ(1) ⊗ ...⊗ vσ(r))
=∑τ∈Sr
sgn(τ)vτσ(1) ⊗ ...⊗ vτσ(r)
= sgn(σ)∑τσ∈Sr
sgn(τσ)vτσ(1) ⊗ ...⊗ vτσ(r)
= sgn(σ)Alt(v1 ⊗ ...⊗ vr)
It follows that f is a constant multiple of Alt.
Let Natformsm (p1, ...pn, q) denote the space of natural operators of type ∧p1T ∗×...×∧pnT ∗ ∧qT ∗
over m-dimensional manifolds. We prove that these spaces are generated (in an operadic sense)
by the wedge product and the exterior derivative.
33
Theorem 4. For all nonnegative integers p1, ..., pn, q, the vector space Natformsm (p1, ..., pn, q) is
spanned by natural operators of form
η : (ω1, ..., ωn) 7→ (∧α1ω1) ∧(∧β1dω1
)∧ ... ∧ (∧αnωn) ∧
(∧βndωn
)where αi and βi are nonnegative integers satisfying
∑ni=1 αipi + βi(pi + 1) = q. If no such αi, βi
exist, f is identically 0.
Proof. Let η : ∧p1T ∗ × ... × ∧pnT ∗ ∧qT ∗ be a natural operator. By theorem 3, η has finite
order r. By theorem 2, η can therefore be identified with a smooth Gr+1m -equivariant map
f :⊕
0≤i≤r1≤j≤n
SiRm∗ ∧pj Rm∗ → ∧qRm∗
Acting by homotheties { 1k id} yields the homogeneity condition
kqf(xi,j) = f(ki+pjxi,j)
By the Homogenous Function Theorem, f decomposes into multi-homogenous components f{di,j}
of degree di,j in SiRm∗ ∧pj Rm∗ with degrees di,j satisfying
q =∑
0≤i≤r1≤j≤n
|di,j |(i+ pj)
If no such di,j exist, f is identically 0. By proposition 8, each multi-homogenous component
f{di,j} is Gr+1m -equivariant. Thus, each f{di,j} corresponds to a linear, Gr+1
m -equivariant map
f{di,j} :⊙
0≤i≤r1≤j≤n
Sdi,j(SiRm∗ ⊗ ∧pjRm∗
)→ ∧qRm∗
Applying lemma 2, we deduce that if di,j > 0 for any i > 1, then f{di,j} is identically 0. Hence,
the only non-zero components are given by linear, Gr+1m -equivariant maps
f{di,j} :
n⊙i=1
Sαi(∧piRm∗)� Sβi(Rm∗ ⊗ ∧piRm∗)→ ∧qRm∗ (15)
34
Since GLm(R) = G1m ⊆ Gr+1
m , f{di,j} is in particular GLm(R)-equivariant. Moreover, both
domain and codomain of (15) are functorial over G-modules. The canonical identification Rm ∼=Rm∗ is an isomorphism of GLm(R)-modules and therefore induces GLm(R)-module isomorphisms
n⊙i=1
Sαi(∧piRm)� Sβi(Rm ⊗ ∧piRm) ∼=n⊙i=1
Sαi(∧piRm∗)� Sβi(Rm∗ ⊗ ∧piRm∗)
∧qRm ∼= ∧qRm∗
Applying these isomorphisms to (15), f{di,j} can be identified with a linear, GLm(R)-equivariant
map
˜f{di,j} :
n⊙i=1
Sαi(∧piRm)� Sβi(Rm ⊗ ∧piRm)→ ∧qRm
replacing Rm with its dual Rm∗. By lemma 2, ˜f{di,j} ,and hence f{di,j}, is a constant multiple of
the alternation. Let us examine how the alternation behaves on different factors of the domain.
Factors like ∧piRm∗ correspond (invariant-theoretically) to 0-jets of differential forms. Since
these factors are already alternating, the alternation behaves like the identity. Factors like
Rm∗⊗∧piRm∗ correspond to 1-jets of differential forms. The alternation maps the 1-jet xi0dxi1∧... ∧ dxip to dxi0 ∧ dxi1 ∧ ... ∧ dxip . Therefore, on these factors the alternation behaves like the
derivative. The multi-homogenous component f{di,j} therefore computes a (scalar multiple of a)
wedge product of pi-forms and their derivatives:
f{di,j}(ω1, ...ωn) = c (∧α1ω1) ∧(∧β1dω1
)∧ ... ∧ (∧αnωn) ∧
(∧βndωn
)c ∈ R
Thus, f is a linear combination of wedge products of forms and their derivatives. Conversely, all
such operators are natural, since both d and ∧ are natural (see examples 7 and 8). So we are
done.
In some cases, the operator
η : (ω1, ..., ωn) 7→ (∧α1ω1) ∧(∧β1dω1
)∧ ... ∧ (∧αnωn) ∧
(∧βndωn
)is identically 0. If |ω| is odd, the skew-symmetry of the wedge product forces ω ∧ ω = 0. If
|ω1| + |ω2| > m, ω1 ∧ ω2 = 0. However, by removing these degeneracies, we obtain a basis for
35
Natm(p1, ..., pn, q).
Proposition 11. Consider the set S of non-zero operators
η : (ω1, ..., ωn) 7→ (∧α1ω1) ∧(∧β1dω1
)∧ ... ∧ (∧αnωn) ∧
(∧βndωn
)for nonnegative αi, βi. The elements of S are linearly independent.
Proof. Denote the operator η : (ω1, ..., ωn) 7→ (∧α1ω1)∧(∧β1dω1
)∧ ...∧ (∧αnωn)∧
(∧βndωn
)by
ηαi,βi . All operators are regarded invariant-theoretically (i.e. as maps∏i T
10 ∧piRm∗ → ∧qRm∗).
Consider a nontrivial linear dependence relation R on some collection of distinct non-zero ηαji ,βji:
R =
t∑j=1
cjηαji ,βji≡ 0
Since ηα1i ,β
1i
is non-zero, there are 1-jets ωi = dxji1 ∧ ... ∧ dxjipi + xki0dxki1 ∧ ... ∧ dxkipi such that
ηα1i ,β
1i(ω1, ..., ωn) 6= 0. For every i and r, s ∈ R, we put
ωi(r, s) = rdxji1 ∧ ... ∧ dxjipi + sxki0dxki1 ∧ ... ∧ dxkipi
Consider the function
f : R2n → ∧qRm∗
(r1, s1, ..., rn, sn) 7→ R(ω1(r1, s1), ..., ωn(rn, sn))
Since d is an R-linear function, f is a polynomial in ri, si. Since ηα1i ,β
1i(ω1, ..., ωn) is non-zero,
it can be completed to a basis. Let π : ∧qRm∗ → R denote the projection onto the linear
subspace Span(ηα1
i ,β1i(ω1, ..., ωn)
)induced by this basis, and let f1 = π ◦ f . By plugging the
forms ω1(r1, s1), ..., ωn(rn, sn) into R and applying the linearity of d, we see that the coefficient
of∏ni=1 r
αii s
βii in f1 is c1, which is non-zero by assumption. Hence, f is non-constant, and
therefore non-zero, a contradiction.
Consider the category ComDga of differential (cohomologically) graded associative supercom-
mutative algebras (commutative dga-algebras, for short) and the forgetful functor F : ComDga→VecZ mapping every commutative dga-algebra to the underlying graded vector space. Consider
36
the functor P : VecZ → ComDga left adjoint to F . P exists and is unique up to natural
isomorphism. For any graded vector space V , we say that P (V ) is the free commutative dga-
algebra generated by V . As a vector space, P (V ) is the free graded associative supercommutative
algebra generated by the graded vector space V ⊕ dV . Note, this is distinct from the notion of
a quasi-free or semi-free dga-algebra, which is free as an associative graded algebra.
Consider the vector space Natformsm spanned by all natural operators of type ∧p1T ∗×...×∧pnT ∗
∧qT ∗ on m-dimensional manifolds, with no restrictions on n, p1, ..., pn, or q. This vector space
is Z-graded by q (as a matter of convention, Natm(p1, ..., pn, q) = 0 for any q < 0).The wedge
product and exterior derivative turn Natformsm into a commutative dga-algebra.
Corollary 4. For every q ∈ N, consider the identity operator idq : ∧qT ∗ ∧qT ∗. Let 〈idq〉 be
the 1-dimensional vector space spanned by idq, and let V be the graded vector space⊕
q∈N〈idq〉.The free commutative dga-algebra P (V ) contains an ideal I>m spanned by elements of degree
exceeding m, the dimension of the base manifolds in question. Natformsm is isomorphic to the
quotient P (V )/I>m.
Proof. Let V =⊕
q∈N〈idq〉. As a vector space, P (V ) is spanned by finite words constructed
from the formal symbols {idq}, a differential d, and an associative, supercommutative degree-0
product ∧, subject to the Leibniz rule:
d(a ∧ b) = da ∧ b+ (−1)|a|a ∧ db
By the universal property, the assignment idq 7→ idq extends to a map of commutative dgas
ϕ : P (V )→ Natformsm , which is surjective by theorem 4. Let x ∈ P (V ) be a homogenous element
of degree q > m. Since ϕ preserves degree, ϕ(x) is a natural operator mapping into smooth
sections of ∧qT ∗. However, over m-dimensional manifolds, the natural bundle ∧qT ∗ is the trivial
bundle M × {0}. Thus, ϕ(x) = 0. It follows that I>m ⊆ kerϕ.
By induction, we see that the non-zero elements
(∧α1idp1) ∧(∧β1d(idp1)
)∧ ... ∧ (∧αnidpn) ∧
(∧βnd(idpn)
)form a basis for P (V ). Suppose x ∈ kerϕ is a linear combination of elements x1, ..., xl as
above. The equation ϕ(c1x1 + ... + clxl) = 0 imposes a nontrivial linear dependence on the
non-zero elements of {ϕ(x1), ..., ϕ(xl)}. But, by proposition 11, these elements are linearly
independent. So, indeed kerϕ ⊆ I>m. Hence, kerϕ = I>m and ϕ descends to an isomorphism
ϕ : P (V )/I>m∼→ Natforms
m .
37
Part II
Natural Operators on Alternating
Multivector Fields
There is a sense in which we got ‘lucky’ in our proof of theorem 4. To classify natural operators on
differential forms it was enough to check equivariance with respect to linear diffeomorphisms (i.e.
elements of the proper subgroup GLm(R) ⊂ Gr+1m ). We are not so lucky in the case of alternating
multivector fields. Recall, from section 3, the group isomorphism Gr+1m∼= NGLr+1
m o GLm(R).
Since the subgroups NGLr+1m and GLm(R) generate Gr+1
m , an operation is natural if and only
if the corresponding map between the standard fibers is equivariant with respect to GLm(R)
and NGLr+1M . In the case of differential forms, the action of NGLr+1
m on GLm(R)-equivariant
operators is manifestly trivial. This is not the case with alternating multivector fields, as demon-
strated below.
Example 22. Consider the bilinear ‘half-bracket’ of vector fields. In local coordinates,
[X,Y ] 12
:=∑i,j
(Xj∂jYi)∂i
For any g ∈ GLm(R), we have (gX)j =∑k g
jkX
k and ∂j(gY )i =∑r,s ∂sY
r(g−1)sjgir. Hence,
38
[gX, gY ] 12
=∑i,j
((gX)j∂j(gY )i)∂i
=∑
i,j,k,r,s
gjkXk∂sY
r(g−1)sjgir∂i
=∑i,k,r,s
Xk∂sYrgir
∑j
(g−1)sjgjk
∂i
=∑i,k,r,s
Xk∂sYrgirδ(s, k)∂i
=∑i,k,r
Xk∂kYrgir∂i
= g[X,Y ] 12
Thus, [X,Y ] 12
is GLm(R)-equivariant. It is not, however, a natural operator. Consider the
invertible 2-jet ϕ : xi 7→ xi + x2i regarded as an element of NGL2
m. Let X =∑i ∂i so that
ϕX =∑i(1 + 2xi)∂i. Clearly, ϕ[X,X] 1
2= ϕ0 = 0 while on the other hand
[ϕX,ϕX] 12
=∑i,j
(ϕX)j∂j(ϕX)i∂i
=∑i
2∂i
= 2X
To classify natural operators on alternating multivector fields, we will adopt the following general
strategy, inspired by the work of Markl [10], [11]:
1. Apply the Invariant Tensor Theorem to model GLm(R)-equivariant operators as linear
combinations of a particular type of multigraph.
2. Define a linear map ∂ on multigraphs that corresponds to the tangent action of ngl∞m on
GLm(R)-equivariant operators.
3. Define a (Lie-admissable) product on multigraphs which behaves suitably with respect to
the ngl∞m -action.
4. Draw analogies to the shuffle coproduct on the tensor algebra to deduce the structure of
ker ∂.
39
9 Twisted Algebras in VecZ
We begin by establishing some of the basic theory of symmetric collections and twisted algebras
over an operad. We deviate slightly from the standard formalism (see, for example, [12]) in that
all vector spaces are graded.
Definition 14. Consider the categories
• FinSet of finite sets with bijections
• VecZ of real, Z-graded vector spaces with linear degree-preserving maps.
Definition 15. A symmetric collection is a functor F : FinSet → VecZ. We write SymCol
for the category of symmetric collections with natural transformations as morphisms.
Thus, a symmetric collection is a family of graded vector spaces {F [S]}, one for each finite set
S, together with linear, degree-preserving maps {F (f) : F [S] → F [S′]}, one for each bijection
f : S → S′ such that F (idS) = idF (S) and F (f ◦ g) = F (f) ◦ F (g) for every pair (f, g) of
composable bijections. A morphism η : F → G of symmetric collections is a family of linear,
degree-preserving maps {ηS : F [S]→ G[S]}, one for each finite set S, such that for every bijection
f : S → S′, the following diagram commutes:
F [S] G[S]
F [S′] G[S′]
ηS
Ff Gf
ηS′
A subobject of F in the category of symmetric collections (a ‘subcollection,’ for lack of a better
term) is a symmetric collection G such that G[S] is contained (as a graded vector space) in F [S],
for every finite set S. Quotient objects are defined pointwise, i.e. (F/G)[S] = F [S]/G[S]. The
classical tensor product of representations has the following analogue in the category of symmet-
ric collections:
Definition 16. For symmetric collections F,G ∈ SymCol, let
(F ⊗G)[S] :=⊕
UtV=S
F [U ]⊗G[V ]
where the tensor product on the right is the ordinary tensor product of graded vector spaces. A
bijection f : S → S′ induces linear isomorphisms F [U ] ∼= F [U ′] and G[V ] ∼= G[V ′] for U ′ = f [U ]
and V ′ = f [V ], and therefore a linear isomorphism (F ⊗G)[S] ∼= (F ⊗G)[S′].
40
Proposition 12. The tensor product of symmetric collections defines a monoidal structure on
SymCol.
Proof. The unit is given by the symmetric collection
1[S] =
{R if S = ∅0 otherwise
Associativity follows from the identity
(F ⊗ (G⊗H))[S] =⊕
UtV tW=S
F [U ]⊗G[V ]⊗H[W ] = ((F ⊗G)⊗H)[S]
we leave the details to the reader.
For every pair of symmetric collections F,G ∈ SymCol, we define the braiding map β(F,G) :
F ⊗G→ G⊗ F by
β(F,G)S : x⊗ y 7→ (−1)|x||y|y ⊗ x
This is the usual switch map in the category of ungraded vector spaces with the standard Koszul
signs. It is easy to see that the braiding maps turn SymCol into a symmetric monoidal category.
Definition 17. An algebra in the category SymCol is a symmetric collection F together with
a morphism × : F ⊗ F → F . Classically, an algebra in the symmetric monodical category of
symmetric collections is called a twisted algebra, a convention we will continue. The morphism
× : F ⊗ F → F is called multiplication on F .
Definition 18. A twisted algebra F with multiplication given by × : F ⊗ F → F is associative
if the following diagram commutes:
F ⊗ F ⊗ F F ⊗ F
F ⊗ F F
×⊗1
1⊗× ×
×
Furthermore, with braiding maps in hand, we can easily define a sensible notion of (super)commutativity.
41
Definition 19. For any symmetric collection F , let S2(F ) denote the subcollection of F ⊗ Fgiven on each finite set U by
S2(F )[U ] = {x ∈ (F ⊗ F )[U ] | β(F, F )U (x) = x}
As in the ungraded case, there is a canonical epimorphism Sym : F ⊗ F → S2(F ) called the
symmetrization map, given by the formula
Sym(x⊗ y) =1
2(x⊗ y + β(F, F )(x⊗ y))
Naturally, we may also consider the nth symmetric power Sn(F ) of a symmetric collection F as
well as the symmetric algebra S(F ), defined as the pointwise sum S(F ) :=⊕
n≥0 Sn(F ).
Definition 20. A twisted supercommutative algebra is a twisted algebra F with multiplication
× : F ⊗ F → F which factors through the symmetric square S2(F ):
F ⊗ F F
S2(F )
×
Sym
One may consider the endomorphism operad of a symmetric collection F in the symmetric
monoidal category SymCol (for a thorough treatment of operads, see, for example, [13]).
Definition 21. For any symmetric collection F , define the endomorphism operad
EndF := {⊗nF → F}n≥0
with the obvious symmetric action and composition law.
This construction suggest an alternative way of defining associative and supercommutative
algebras—indeed, algebras of any desired type—in the context of symmetric collections.
Definition 22. Let O be an operad. A twisted O-algebra is a symmetric collection F together
with an operad map O → EndF .
A twisted associative algebra is therefore a symmetric collection F together with an operad map
Ass → EndF where Ass denotes the operad governing graded associative algebras. A twisted
associative, supercommutative algebra is a symmetric collection F together with an operad map
Com→ EndF where Com denotes the operad governing associative, supercommutative algebras.
42
We conclude this section by presenting an alternative (equivalent) definition of the category of
symmetric collections in terms of sequences of (graded) representations:
Sequence Definition of Symmetric Collection. An object in SymCol is a sequence of
graded vector spaces V = {Vn}n≥0, where each Vn is a graded representation of the symmetric
group Sn. A morphism f : V → W is a sequence {fn : Vn → Wn}n≥0, where each fn is a
morphism of Sn-representations.
Notation. For each n, the subspace of homogenous degree-k elements in Vn is denoted by V kn
so that
Vn =⊕k∈Z
V kn
‘Graded representation of the symmetric group’ in the definition above means that this a direct
sum of Sn-representations.
The functor F 7→ {F [n]}n≥0 defines an obvious equivalence between our two definitions. The
tensor product of functors F : FinSet→ VecZ in definition 16 becomes:
(V ⊗W )n =⊕i+j=n
IndSnSi×Sj (Vi ⊗Wj)
where Ind denotes the induced representation. Concretely, each summand IndSnSi×Sj (Vi ⊗Wj) is
given by
IndSnSi×Sj (Vi ⊗Wj) =⊕
(i,j)−shuffles σ
σ Vi ⊗Wj
That is, a direct sum of(ni
)isomorphic copies of Vi⊗Vj , indexed by (i, j)-shuffles. Fix a particular
(i, j)-shuffle σ. Since the (i, j)-shuffles form a complete set of coset representatives for the normal
subgroup Si×Sj ⊂ Sn, for each permutation g ∈ Sn, there is a unique element xy ∈ Si×Sj and
(i, j)-shuffle τ such that gσ = τxy. The element g then acts via
g(σ v ⊗ w) = τ x(v)⊗ y(w)
The braiding map β(V,W ) : V ⊗W →W ⊗ V becomes:
43
β(V,W )n(σ v ⊗ w) = (−1)|v||w|(σρi,j) w ⊗ v
where ρi,j is the (i, j)-shuffle switching the first i and last j elements of {1, ..., n}. We have the
following characterization of twisted associative algebras in the sequence model for symmetric
collections:
Proposition 13. A twisted associative algebra is a symmetric collection {Vn}n≥0 together with
a family of linear, degree-preserving, Si × Sj-equivariant maps ϕi,j : Vi ⊗ Vj → Vi+j such that
the following diagram commutes:
Vi ⊗ Vj ⊗ Vk Vi+j ⊗ Vk
Vi ⊗ Vj+k Vi+j+k
ϕi,j⊗1
1⊗ϕj,k ϕi+j,k
ϕi,j+k
for every i, j, k ∈ N.
Proof. Let V = {Vn}n≥0 be a twisted associative algebra with multiplication given by the mor-
phism f : V ⊗ V → V . For each n, we have a map of graded Sn-representations:
fn :⊕i+j=n
IndSnSi×Sj (Vi ⊗ Vj) = (V ⊗ V )n → Vn
By the discussion above, the tensor product Vi ⊗ Vj is contained as a Si × Sj-subrepresentation
in (V ⊗ V )n. Right-composing with the canonical inclusion i : Vi ⊗ Vj ↪→ (V ⊗ V )n, we obtain a
map of graded Si × Sj-representations
fn ◦ i : Vi ⊗ Vj → Vn
which we will denote by the symbol ϕi,j . Now, let i, j, and k be arbitrary natural numbers and
consider elements vi ∈ Vi, vj ∈ Vj , and vk ∈ Vk. Since V is twisted associative, we have
ϕi+j,k(ϕi,j(v1 ⊗ v2)⊗ v3) = f(f(vi ⊗ vj)⊗ vk)
= (f)(f ⊗ 1)(vi ⊗ vj ⊗ vk)
= (f)(1⊗ f)(vi ⊗ vj ⊗ vk)
= f(vi ⊗ f(vj ⊗ vk))
= ϕi,j+k(vi ⊗ ϕj,k(vj ⊗ vk))
44
as desired. Conversely, suppose V is a symmetric collection with a family of linear, degree-
preserving, Si×Sj-equivariant maps ϕi,j : Vi⊗Vj → Vi+j such that the diagram in the proposition
commutes. For each n, define an Sn-equivariant map fn :⊕
i+j=n IndSnSi×Sj (Vi ⊗ Vj) = (V ⊗V )n → Vn by putting
fn(σ vi ⊗ vj) = σ ϕi,j(vi ⊗ vj)
and extending by linearity. The collection of maps f = {fn}n∈N defines a morphism V ⊗V → V .
We want to show that (V, f) is a twisted associative algebra. An element of the tensor cube
V ⊗ V ⊗ V is of form σ vi ⊗ vj ⊗ vk where σ ∈ Si+j+kSi×Sj×Sk is an arbitrary (i, j, k)-shuffle. The
groupSi+j+k
Si×Sj×Sk of (i, j, k) shuffles contains normal subgroupsSi+j+kSi+j×Sk and
Si+jSi×Sj with trivial
intersection. Hence, the commutator subgroup
[Si+j+kSi+j × Sk
,Si+jSi × Sj
] ⊆ Si+j+kSi+j × Sk
∩ Si+jSi × Sj
= 1
and these two subgroups commute. The multiplication map
Si+j+kSi+j × Sk
× Si+jSi × Sj
→ Si+j+kSi × Sj × Sk
is therefore a group homomorphism with trivial kernel. By simple counting, it follows that
the multiplication map is surjective and, hence, a group isomorphism. Similarly, we see that
multiplication defines a group isomorphism
Si+j+kSi × Sj+k
× Sj+kSj × Sk
∼=Si+j+k
Si × Sj × Sk
The upshot, of course, is that each (i, j, k)-shuffle σ decomposes uniquely as a commutative
product σi+j,kσi,j of an(i + j, k)-shuffle and an (i, j)-shuffle, as well as a commutative product
σj,kσi,j+k of an (i, j + k)-shuffle and a (j, k)-shuffle. The associativity of f follows trivially,
45
(f)(f ⊗ 1)(σ vi ⊗ vj ⊗ vk) = fi+j+k(σi+j,k fi+j(σi,j vi ⊗ vj)⊗ vk)
= σi+j,k fi+j+k(σi,j fi+j(vi ⊗ vj)⊗ vk)
= σi+j,k ϕi+j,k(σi,j ϕi,j(vi ⊗ vj)⊗ vk)
= σi+j,kσi,j ϕi+j,k(ϕi,j(vi ⊗ vj)⊗ vk)
= σ ϕi,j+k(vi ⊗ ϕj,k(vj ⊗ vk))
= σi,j+kσj,k fi+j+k(vi ⊗ fj+k(vj ⊗ vk))
= fi+j+k(σi,j+k vi ⊗ σj,k fj+k(vj ⊗ vk))
= fi+j+k(σi,j+k vi ⊗ fj+k(σj,k vj ⊗ vk))
= (f)(1⊗ f)(σ vi ⊗ vj ⊗ vk)
Quite similarly, we obtain the following useful characterization of twisted supercommutative al-
gebras:
Proposition 14. A twisted supercommutative algebra is a symmetric collection {Vn}n≥0 to-
gether with a family of linear, degree-preserving, Si × Sj-equivariant maps ϕi,j : Vi ⊗ Vj → Vi+j
such that ϕi,j(v1 ⊗ v2) = (−1)|v1||v2|ρi,jϕj,i(v2 ⊗ v1).
Proof. Let V = {Vn}n≥0 be a twisted supercommutative algebra with multiplication given by
the morphism f : V ⊗ V → V . For each n, we have a map of graded Sn-representations:
fn :⊕i+j=n
IndSnSi×Sj (Vi ⊗ Vj) = (V ⊗ V )n → Vn
and the induced map of graded Si × Sj-representations
ϕi,j := fn ◦ i : Vi ⊗ Vj → Vn
Since V is twisted supercommutative, we have the following relation for every vi ∈ Vi and vj ∈ Vj :
46
ϕi,j(vi ⊗ vj) = fn(vi ⊗ vj)
= fn (β(V, V )n(vi ⊗ vj))
= fn
((−1)|vi||vj |ρi,j vj ⊗ vi
)= (−1)|vi||vj |ρi,jfn(vj ⊗ vi)
= (−1)|vi||vj |ρi,jϕj,i(vj ⊗ vi)
as desired. Conversely, suppose V is a symmetric collection with linear, degree-preserving, Si×Sj-equivariant maps ϕi,j : Vi⊗Vj → Vi+j satisfying the condition ϕi,j(v1⊗v2) = (−1)|v1||v2|ρi,jϕj,i(v2⊗v1). For each n, define an Sn-equivariant map fn :
⊕i+j=n IndSnSi×Sj (Vi ⊗ Vj) = (V ⊗ V )n → Vn
by putting
fn(σ vi ⊗ vj) = σ ϕi,j(vi ⊗ vj)
and extending by linearity. The collection of maps f = {fn}n∈N defines a morphism V ⊗V → V .
We want to show that (V, f) is a twisted supercommutative algebra. Indeed, since ρi,jρj,i = 1,
f (β(V, V )n(σ vi ⊗ vj)) = fi+j (β(V, V )n(σ vi ⊗ vj))
= fi+j
((−1)|v1||v2|(σρi,j) vj ⊗ vi
)= (−1)|v1||v2|(σρi,j) fi+j(vj ⊗ vi)
= (−1)|v1||v2|(σρi,j) ϕj,i(vj ⊗ vi)
= (−1)|v1||v2|(−1)|v2||v1|(σρi,jρj,i) ϕi,j(vi ⊗ vj)
= σ ϕi,j(vi ⊗ vj)
= fi+j(σ vi ⊗ vj)
Let τ be a simple transposition on three letters with τ(l) = l + 1, and let τi,j,k ∈ Si+j+k be the
(i, j, k)-shuffle determined by τ . For example, if τ is (12), τi,j,k switches 1, ..., i with i+1, ..., i+j.
If τ is (23), τi,j,k switches i+1, ..., j with i+j+1, ..., i+j+k. Let σ v1⊗v2⊗v3 ∈ IndSi+j+kSi×Sj×Sk(Vi⊗
Vj ⊗ Vk) ⊂ ⊗3V be an arbitrary element of the tensor cube. τ acts on σ v1 ⊗ v2 ⊗ v3 via
τ (σ v1 ⊗ v2 ⊗ v3) = −(−1)(|vl|−1)(|vl+1|−1)στi,j,k vτ(1) ⊗ vτ(2) ⊗ vτ(3)
47
This extends (uniquely) to a group action of S3 on the tensor cube ⊗3V . The fixed points form
a subcollection of ⊗3V , which we will denote by A3(V ) (notice the difference between A3(V )
and S3(V )). Consider the epimorphism ΦV : ⊗3V → A3(V ) defined by
ΦV (σ v1 ⊗ v2 ⊗ v3) =∑τ∈S3
1
6τ(σ v1 ⊗ v2 ⊗ v3)
Definition 23. A twisted Lie-admissable algebra is a symmetric collection V together with a
morphism f : V ⊗ V → V such that
(f(f ⊗ 1)− f(1⊗ f)) ΦV = 0
One easily sees that in arity 0, definition 23 coincides with the usual definition of Lie-admissability.
In many ways, the sequence definition is better suited for the particular context at hand. In
what follows a ‘symmetric collection’ will almost always refer to a sequence of graded represen-
tations.
10 Ordered Multigraphs
An ordered multigraph is a directed multigraph with total orderings on half-edges. Formally,
Definition 24. An ordered multigraph is a sextuple (V,E, s, t, <s, <t). V is a finite set of ver-
tices, E is a finite set of edges, s : E → V is the source function, t : E → V is the target function,
and <s and <t are functions which assign total orderings on s−1(v) and t−1(v), respectively, to
every vertex v ∈ V .
Definition 25. A morphism of ordered multigraphs is a pair of functions (f : V1 → V2, g : E1 →E2) such that the following diagrams commute:
E1 E2
V1 V2
g
s1 s2
f
E1 E2
V1 V2
g
t1 t2
f
and such that the edge function g : E1 → E2 is monotonic with respect to all relevant orderings.
When drawing these objects, we will adopt the (arbitrary) convention of ordering inputs and
outputs left to right on the page. When in doubt, all multigraphs in this and subsequent sec-
tions are ordered. Multigraphs may be labeled or unlabeled. A ‘rooted’ multigraph is a labeled,
48
ordered multigraph with a special marked vertex (its root) with no outgoing edges. For example
•x
�
a b
is a simple rooted multigraph. The vertex set is {x,�}, with � denoting the root, the edge set
is {a, b}, the source function is identically x, and the target function is identically �. The inputs
to � are ordered by a <t b and the outputs from x are ordered by a <s b. In what follows, an
‘ordinary’ vertex in a rooted multigraph is any vertex distinct from the root.
Definition 26. For every natural number n, let Gra[n] denote the real vector space spanned by
rooted multigraphs with ordinary vertices labeled 1, ..., n, such that every ordinary vertex has at
least one outgoing edge. Each Gra[n] is a Z-graded vector space, with the degree of a multigraph
defined as the valence of its root.
For each n, Sn acts on the graded vector space Gra[n] by relabeling.
Definition 27. Let Gra denote the symmetric collection {Gra[n]}n≥0.
Many of the multigraphs we will consider are contained in one of the following two subcollections:
Definition 28. A rooted multigraph is irreducible if the multigraph formed by removing the
root is connected. Grairred is the subcollection of irreducible multigraphs (i.e. Grairred[n] is the
subspace of Gra[n] spanned by irreducible multigraphs, for every n ∈ N).
Definition 29. A rooted multigraph is internally acyclic if the multigraph formed by removing
the root is acyclic. Graacyc is the subcollection of internally acyclic multigraphs (i.e. Graacyc[n]
is the subspace of Gra[n] spanned by internally acyclic multigraphs, for every n ∈ N).
Remark 4. It is worth noting that every multigraph G ∈ Graacyc is necessarily connected.
Suppose G is disconnected and select a connected component G′ ⊂ G not containing the root.
Since G ∈ Graacyc, G′ is acyclic and therefore, by simple order theory, contains a maximal
element, i.e. a vertex with no outgoing edges. However, the only such vertex permitted in Gra
is the root, which, by assumption, is not contained in G′.
We will often work in the intersection Grairred ∩Graacyc, which we denote, somewhat unimagi-
49
natively, by Grairredacyc . Finally,
Definition 30. We say that a rooted multigraph G ∈ Gra[n] is of type (α1, ..., αn) if for each
i ∈ [n], the vertex labeled i has precisely αi outgoing edges. Gra(α1, ..., αn) ⊂ Gra[n] is the
subspace spanned by type-(α1, ..., αn) multigraphs.
The symmetric collection Gra has the structure of a twisted associative algebra. The multipli-
cation maps ∧ : Gra[i]⊗Gra[j]→ Gra[n] are defined as follows:
Definition 31. Let G1 ∈ Gra[i] and G2 ∈ Gra[j]. Construct the multigraph G1∧G2 ∈ Gra[i+j]
by
1. Relabeling each vertex k ∈ G2 by the natural number i+ k
2. Removing the root vertex of G2
3. Grafting the free inputs from the root vertex of G2 onto the root vertex of G1
The inputs to the root vertex of the wedge product G1 ∧G2 are ordered by concatenation. All
other orderings on G1 ∧G2 are inherited from G1 or G2 in the obvious way. Finally, this map is
extended to the tensor product Gra[i]⊗Gra[j] by bilinearity.
Example 23.
•1
�
∧
•1
�
=
•1
•2
�
We would like to introduce a certain collection of symmetries and anti-symmetries on the half-
edges of our multigraphs. This can be accomplished by modding out by a suitable subcollection.
Definition 32. Let Z ⊂ Gra be the subcollection spanned by elements of the following three
types
50
�
· · · +
�
· · ·
Type 1
•
· · ·
· · ·
−•
· · ·
· · ·
Type 2
•
· · ·
· · ·
+
•
· · ·
· · ·
Type 3
Note that crossings in the pictures above represent arbitrary transpositions of the indicated type.
Definition 33. Let Gra denote the quotient Gra/Z and π : Gra � Gra the canonical epi-
morphism. Denote the images of the subcollections Grairred, Graacyc, and Grairredacyc under π by
Grairred
, Graacyc, and Grairred
acyc , respectively.
We regard elements of the spaces Gra[n] as rooted multigraphs labeled by [n] with
1. Symmetric inputs to all ordinary vertices
2. Anti-symmetric ouputs
3. Anti-symmetric inputs to the root
The wedge product ∧ : Gra ⊗ Gra → Gra induces a product on Gra, denoted by the same
symbol, in the obvious way:
51
∧ : Gra[i]⊗Gra[j]→ Gra[n]
G1 ⊗G2 7→ G1 ∧G2
It is a simple exercise to verify that these maps are well-defined.
Proposition 15. The wedge product turns Gra into a twisted associative, super-commutative
algebra, freely generated over Grairred
.
Proof. Associativity is obvious. Let G1 ∈ Gra[i] and G2 ∈ Gra[j]. By proposition 14, we need
to show that G1 ∧ G2 = (−1)|G1||G2|ρi,j G2 ∧ G1. Reflecting on the definition of the wedge
product, it is clear that the multigraph G1 ∧G2 can be transformed into G2 ∧G1 by performing
|G1||G2| transpositions on inputs to the root and then applying ρi,j to the vertices of the resulting
multigraph. In Gra[i+ j], each transposition comes with a sign of −1. Hence,
G1 ∧G2 = G1 ∧G2 = (−1)|G1||G2|ρi,j G2 ∧G1 = (−1)|G1||G2|ρi,j G1 ∧G2
Moreover, each rooted multigraph G decomposes uniquely (up to re-ording) as a wedge product
of irreducible pieces. So indeed, Gra ∼= S(Gra
irred)
, as desired.
11 Ordered Multigraphs and Natural Operators
Rooted multigraphs correspond quite nicely to certain spaces of GLm(R)-invariant tensors, which
arise organically in the study of natural operators on alternating multivector fields. Let V = Rm.
For natural numbers α1, ..., αn, γ consider the GL(V )-module
T(α1, ..., αn)(γ) :=⊕
β1,...,βn∈N
n⊗i=1
(⊗βiV ⊗⊗αiV ∗
)⊗⊗γV
and the submodule
W(α1, ..., αn)(γ) :=⊕
β1,...,βn∈N
n⊗i=1
(SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
Put
52
T(α1, ..., αn) :=⊕γ∈N
T(α1, ..., αn)(γ) W(α1, ..., αn) :=⊕γ∈N
W(α1, ..., αn)(γ)
We regard each as a Z-graded vector space, with degree defined by γ.
Definition 34. Let T and W be the symmetric collections defined by
T[n] :=⊕
α1,...,αn∈NT(α1, ..., αn) W[n] :=
⊕α1,...,αn∈N
W(α1, ..., αn)
Sn acts on components by permuting factors.
The inclusion i : W ↪→ T admits a GL(V )-equivariant left-inverse p : T � W given on each
component by the formula:
pn :
n⊗i=1
(vi1⊗...⊗viβi⊗v∗i1⊗...⊗v
∗iαi
)⊗v1⊗...⊗vγ 7→n⊗i=1
Sym((vi1⊗...⊗viβi )⊗Alt(v∗i1⊗...⊗v
∗iαi
)⊗Alt(v1⊗...⊗vγ)
We denote the subcollections of GL(V )-invariants in the standard way by
TGL(V )[n] := T[n]GL(V ) WGL(V )[n] := W[n]GL(V )
Since p |TGL(V ) ◦ i |WGL(V )= id, p maps TGL(V ) surjectively onto WGL(V ).
Analogously, let Natfieldsm (α1, ..., αn)(γ) denote the vector space of natural multilinear operators
η : ∧α1T × ...× ∧αnT ∧γT over m-dimensional manifolds. Put
Natfieldsm (α1, ..., αn) :=
⊕γ∈N
Natfieldsm (α1, ..., αn)(γ)
grading, as before, by γ.
Definition 35. Let Natfieldsm be the symmetric collection defined by
Natfieldsm [n] =
⊕α1,...,αn∈N
Natfieldsm (α1, ..., αn)
Sn acts by permuting arguments.
53
The connection between natural multilinear operators on alternating multivector fields and in-
variant tensors in W is expressed in the following simple proposition.
Proposition 16. For every α1, ..., αn ∈ N, restriction to the standard fiber defines a linear
isomorphism
Natfieldsm (α1, ..., αn) ∼=
(W(α1, ..., αn)GL(V )
)NGL∞mThese isomorphisms assemble into an isomorphism of symmetric collections:
Natfieldsm
∼=(WGL(V )
)NGL∞mProof. Consider the subspace Natfields
m,r (α1, ..., αn)(γ) ⊂ Natfieldsm (α1, ..., αn)(γ) consisting of nat-
ural multilinear operators η : ∧α1T × ...× ∧αnT ∧γT of order not exceeding r. By invariant-
theoretic reduction and remark 3, the restriction defines a linear isomorphism
Natfieldsm,r (α1, ..., αn)(γ) ∼=
((n⊗i=1
T rm ∧αi V ∗)⊗ ∧γV
)Gr+1m
We have the natural identifications
(n⊗i=1
T rm ∧αi V ∗)⊗ ∧γV =
n⊗i=1
r⊕j=0
SjV ⊗ ∧αiV ∗⊗ ∧γV
=⊕
(β1,...,βn)∈{0,...,r}n
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)
so that
54
Natfieldsm (α1, ..., αn)(γ) ∼=
∞⋃r=0
⊕(β1,...,βn)∈{0,...,r}n
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)Gr+1m
=
∞⋃r=0
⊕(β1,...,βn)∈{0,...,r}n
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)G∞m
=
∞⋃r=0
⊕(β1,...,βn)∈{0,...,r}n
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)G∞m
=
⊕β1,...,βn∈N
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)G∞m
and
Natfieldsm (α1, ..., αn) =
⊕γ∈N
Natfieldsm (α1, ..., αn)(γ)
∼=⊕γ∈N
⊕β1,...,βn∈N
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)G∞m
=
⊕β1,...,βn,γ∈N
n⊗i=1
((SβiV ⊗ ∧αiV ∗
)⊗ ∧γV
)G∞m
= W(α1, ..., αn)G∞m
Recall, Gr+1m is isomorphic to a semidirect product NGLr+1
m oGLm(R). Taking the inverse limit,
we obtain, G∞m∼= NGLr+1
m o GLm(R). In particular, the subgroups NGL∞m , GLm(R) ⊆ G∞m
generate G∞m . Hence,
Natfieldsm (α1, ..., αn) ∼=
(W(α1, ..., αn)GL(V )
)NGL∞mwhich establishes the first part. Since Natfields
m [n] =⊕
α1,...,αn∈NNatfieldsm (α1, ..., αn) is a direct
sum of G∞m -modules, these isomorphisms assemble:
Natfieldsm [n] =
⊕α1,...,αn∈N
Natfieldsm (α1, ..., αn) ∼=
⊕α1,...,αn∈N
W(α1, ..., αn)G∞m = WG∞m =
(WGL(V )
)NGL∞m55
Each isomorphism is (trivially) an isomorphism of Sn-representations, since permuting the ar-
guments of a natural operation permutes the factors in the standard fiber of the corresponding
jet bundle.
We wish to define a correspondence between elements of the symmetric collection Gra and
WGL(V ). Every multigraph G ∈ Gra[n] gives rise to an elementary invariant tensor fn(G) ∈TGL(V )[n] in the following way: the multigraph G has n ordinary vertices labeled 1, ..., n. For
each i ∈ [n], let αi, βi denote the number of outgoing and incoming edges, respectively, to the
ith vertex of G and let γ denote the number of incoming edges to the root. The structure of G
determines a bijection σ(g) from inputs onto outputs, which determines an elementary invariant
tensor Iσ(G) ∈⊗n
i=1
((⊗βiV ⊗⊗αiV ∗
)⊗⊗γV
). We put fn(G) = Iσ(g) and extend to Gra[n]
by linearity. The morphism f : Gra → TGL(V ) is surjective, by the Invariant Tensor Theo-
rem. Moreover, if m exceeds∑ni=1 αi, we have ker fn ∩ Gra(α1, ..., αn) = 0. Fix an inclusion
i : Gra ↪→ Gra right inverse to π.
Definition 36. Let f : Gra→WGL(V ) be the morphism of symmetric collections given by the
composition f =: p ◦ f ◦ i
The following diagram commutes:
Gra TGL(V )
Gra WGL(V )
f
π p
f
i i
We see immediately that
Proposition 17. f
1. is well-defined (i.e. independent of i)
2. is surjective
3. restricts to a linear isomorphism fn : Gra(α1, ..., αn)∼→W(α1, ..., α)GL(V ) for
∑ni=1 αi < m
Proof. All three claims follow from the trivial observation f(Z) = ker p.
1. If i1 and i2 are two right-inverses of π and g ∈ Gra, we have π(i1g − i2g) = g − g = 0.
Hence, i1g− i2g ∈ kerπ = Z and therefore, since f(Z) = ker p, (p ◦ f ◦ i1 − p ◦ f ◦ i2)(g) =
p(f((i1g − i2g))) = 0, as desired.
56
2. Since f is surjective, it admits a right-inverse g. We claim π ◦ g ◦ i is a right-inverse for f .
Let x ∈ TGL(V ). Notice that (i ◦π ◦ g)(x)− g(x) ∈ kerπ since π((i ◦π ◦ g)(x)− g(x)) = (π ◦g)(x)−(π◦g)(x) = 0. Therefore, since f(Z) = ker p, we have (p◦f)((i◦π◦g)(x)−g(x)) = 0
and hence (p◦f ◦ i◦π ◦g)(x) = (p◦f ◦g)(x) = p(x). Now, letting x = i(y) for y ∈WGL(V ),
we see that (p ◦ f ◦ i ◦π ◦ g ◦ i)(y) = (p ◦ i)(y) = y. However, p ◦ f ◦ i ◦π ◦ g ◦ i = f ◦π ◦ g ◦ iso, indeed, f ◦ π ◦ g ◦ i = id, as desired.
3. Let α1, ..., αn be natural numbers with∑ni=1 αi < m. Let f ′, f
′, π′, and p′ denote
the restrictions of f , f , π, and p to Gra(α1, ..., αn), Gra(α1, ..., αn), Gra(α1, .., αn), and
T(α1, ..., αn)GL(V ), respectively. Note that the following diagram commutes
Gra(α1, ..., αn) T(α1, ..., αn)GL(V )
Gra(α1, ..., αn) W(α1, ..., αn)GL(V )
f ′
p′
f′
i
The top row is an isomorphism, by the Invariant Tensor Theorem. And fn is diagonal with
respect to the decomposition Gra[n] =⊕Gra(α1, ..., αn). Thus, f(Z) = ker p implies
f(kerπ′) = ker p′. Suppose x ∈ ker f′
so that (p′ ◦f ′ ◦ i)(x) = 0. It follows that (f ′ ◦ i)(x) ∈ker p′ = f ′(kerπ′). In other words, there is a y ∈ kerπ′ such that f ′(i(x)−y) = (f ′ ◦ i)(x)−f ′(y) = 0. Notice that π′(i(x) − y) = x − π′y = x and, clearly, i(x) − y ∈ ker f ′. Hence,
x ∈ π′ ker f ′. Thus, we have proved ker f′ ⊆ π′ ker f ′ = 0. f
′is surjective onto WGL(V ) by
part 2, so we are done.
In summary, we have proved
1. Natural multilinear operators on alternating multivector fields correspond to certain types
of invariant tensors
2. Invariant tensors correspond (in stable dimension) to certain spaces of rooted multigraphs
12 Gerstenhaber Algebras
A Gerstenhaber algebra is a graded associative, supercommutative algebra with a lie bracket of
degree −1 satisfying the poisson identity. More formally
Definition 37. A Gerstenhaber algebra is a graded vector space G =⊕∞
i=1Gi equipped with a
wedge product ∧ : G⊗G→ G and a lie bracket [ , ] : G⊗G→ G. The degree of a homogenous
element a is denoted by |a|, and the following identities hold:
57
1. |a ∧ b| = |a|+ |b| (the wedge product has degree 0)
2. |[a, b]| = |a|+ |b| − 1 (the lie bracket has degree −1)
3. a ∧ (b ∧ c) = (a ∧ b) ∧ c (the wedge product is associative)
4. [a, [b, c]] = [[a, b], c] + (−1)(|a|−1)(|b|−1)[b, [a, c]] (the Jacobi Identity)
5. a ∧ b = (−1)|a||b|b ∧ a (the wedge product is supercommutative)
6. [a, b] = −(−1)(|a|−1)(|b|−1)[b, a] (the lie bracket is (graded) antisymmetric)
7. [a, b ∧ c] = [a, b] ∧ c+ (−1)(|a|−1)|b|b ∧ [a, c] (the Poisson identity)
Example 24. The Hochschild cohomology H∗(A,A) of an associative algebra A is a Gersten-
haber algebra under the cup product and a lie bracket defined by Gerstenhaber [14].
Example 25. The graded vector space of alternating multivector fields forms a Gerstenhaber
algebra under the wedge product of multivector fields and the Schouten-Nijenhuis bracket.
Example 26. The vector space of natural multilinear operations on alternating multivector
fields, graded by the degree of the output, forms a Gerstenhaber algebra under the wedge prod-
uct and Schouten-Nijenhuis bracket.
Example 27. The homology of the little-disks operad is the operad for Gerstenhaber algebras,
a result first proved in [15].
13 Pre-Gerstenhaber Algebras
We will now formalize the notion of a pre-Gerstenhaber algebra. Roughly speaking, pre-Gerstenhaber
algebras are to Gerstenhaber algebras as Lie-admissable algebras are to Lie Algebras. Formally,
Definition 38. A pre-Gerstenhaber algebra is a graded vector space P =⊕∞
i=1 Pi together with
two bilinear multiplications ∧, ∗ : P ⊗ P → P such that the following identities hold:
1. |a ∧ b| = |a|+ |b| (the wedge product has degree 0)
2. |a ∗ b| = |a|+ |b| − 1 (the asterisk has degree −1)
3. a ∧ (b ∧ c) = (a ∧ b) ∧ c (the wedge product is associative)
4. a ∧ b = (−1)|a||b|b ∧ a (the wedge product is supercommutative)
58
5. a ∗ (b ∧ c) = (a ∗ b) ∧ c + (−1)(|a|−1)|b|b ∧ (a ∗ c) (the asterisk is a left-derivation over the
wedge product)
6. (a ∧ b) ∗ c = a ∧ (b ∗ c) + (−1)(|c|−1)|b|(a ∗ c) ∧ b (the asterisk is a right-derivation over the
wedge product)
Furthermore, if [a, b] = a ∗ b − (−1)(|a|−1)(|b|−1)b ∗ a, we have the following Lie-admissability
condition
7. [a, [b, c]] = [[a, b], c] + (−1)(|a|−1)(|b|−1)[b, [a, c]] (the asterisk is Lie-admissable)
Let A be a graded, Lie-admissable algebra, and consider the pre-Gerstenhaber algebra P (A)
generated freely over A (more precisely, consider the functor P : LieAdm → Pre-Gerst left
adjoint to the forgetful functor F : Pre-Gerst → LieAdm and apply P to A). What can we
say, if anything, about the structure of P (A)? As a vector space, P (A) is a quotient of the free
graded supercommutative associative algebra generated by the graded vector space A. Naively,
one might expect a rather unwieldy set of relations in the denominator of this quotient, imposed
by the interaction between the Poisson identities and the Lie-admissability condition. This is,
remarkably, not the case.
Theorem 5. Let A be a graded, Lie-admissable algebra and P (A) the free pre-Gerstenhaber
algebra generated by A. P (A) is isomorphic as a vector space (and, indeed, as a graded algebra)
to the free graded supercommutative associative algebra generated by the graded vector space
A.
Proof. We must demonstrate the consistency of pre-Gerstenhaber axioms 5, 6, and 7 on the
free graded supercommutative associative algebra generated by the graded vector space V . This
object can be constructed by
1. forming the tensor algebra over A (regarded as an ungraded vector space)
2. grading in accordance with the given grading on A and the formula |a ∧ b| = |a|+ |b|
3. modding out by supercommutativity
To check the consistency of axioms 5 and 6, we compute the product (a ∧ b) ∗ (c ∧ d) in two
different ways. Applying the left Poisson relation first, we obtain:
59
(a ∧ b) ∗ (c ∧ d) = ((a ∧ b) ∗ c) ∧ d+ (−1)(|a|+|b|−1)|c|c ∧ ((a ∧ b) ∗ d)
= a ∧ (b ∗ c) ∧ d+ (−1)(|c|−1)|b|(a ∗ c) ∧ b ∧ d
+ (−1)(|a|+|b|−1)|c|(c ∧ a ∧ (b ∗ d) + (−1)(|d|−1)|b|c ∧ (a ∗ d) ∧ b)
= a ∧ (b ∗ c) ∧ d+ (−1)(|c|−1)|b|(a ∗ c) ∧ b ∧ d
+ (−1)(|a|+|b|−1)|c|c ∧ a ∧ (b ∗ d) + (−1)(|a|+|b|−1)|c|+(|d|−1)|b|c ∧ (a ∗ d) ∧ b
Applying the right Poisson relation first, we obtain:
(a ∧ b) ∗ (c ∧ d) = a ∧ (b ∗ (c ∧ d)) + (−1)(|c|+|d|−1)|b|(a ∗ (c ∧ d)) ∧ b
= a ∧ (b ∗ c) ∧ d+ (−1)(|c|−1)|d|a ∧ c ∧ (b ∗ d)
+ (−1)(|c|+|d|−1)|b|((a ∗ c) ∧ d ∧ b+ (−1)(|a|−1)|c|c ∧ (a ∗ d) ∧ b)
= a ∧ (b ∗ c) ∧ d+ (−1)(|c|−1)|d|a ∧ c ∧ (b ∗ d)
+ (−1)(|c|+|d|−1)|b|(a ∗ c) ∧ d ∧ b+ (−1)(|c|+|d|−1)|b|+(|a|−1)|c|c ∧ (a ∗ d) ∧ b
The first and fourth terms in both expansions are the same. The second term in the first
expansion is equal to the third term in the second (here, applying the supercommutativity of the
wedge product) and the third term in the first expansion is equal to the second in the second.
However, this part of the proof is hardly surprising. Clearly, the consistency of axioms 6 and
7 follows from the consistency of the right and left Poisson relations on the free Gerstenhaber
algebra over A.
The interesting observation is that if we assume the Poisson relations, we get the Jacobi identity
for free. We know that the Jacobi identity is satisfied by all triples in A. Thus, by induction, it
suffices to prove the following: if the triples (a, x, y) and (b, x, y) satisfy the Jacobi identity, so
does the triple (a ∧ b, x, y). For notational convenience, we will omit the absolute values when
writing degrees:
60
[a ∧ b, [x, y]] = (a ∧ b) ∗ (x ∗ y − (−1)(x−1)(y−1)y ∗ x)− (−1)(a+b−1)(x+y)(x ∗ y − (−1)(x−1)(y−1)y ∗ x) ∗ (a ∧ b)
= a ∧ (b ∗ (x ∗ y)) + (−1)(x+y)b(a ∗ (x ∗ y)) ∧ b
− (−1)(x−1)(y−1)a ∧ (b ∗ (y ∗ x))− (−1)(x+y)b+(x−1)(y−1)(a ∗ (y ∗ x)) ∧ b
− (−1)(a+b−1)(x+y)((x ∗ y) ∗ a) ∧ b− (−1)(b−1)(x+y)a ∧ ((x ∗ y) ∗ b)
+ (−1)(a+b−1)(x+y)+(x−1)(y−1)((y ∗ x) ∗ a) ∧ b+ (−1)(b−1)(x+y)+(x−1)(y−1)a ∧ ((y ∗ x) ∗ b)
[[a ∧ b, x], y] = ((a ∧ b) ∗ x− (−1)(a+b−1)(x−1)x ∗ (a ∧ b)) ∗ y
− (−1)(a+b+x)(y−1)y ∗ ((a ∧ b) ∗ x− (−1)(a+b−1)(x−1)x ∗ (a ∧ b))
= a ∧ ((b ∗ x) ∗ y) + (−1)(y−1)(b+x−1)(a ∗ y) ∧ (b ∗ x)
+ (−1)(x−1)b(a ∗ x) ∧ (b ∗ y) + (−1)(x+y)b((a ∗ x) ∗ y) ∧ b
− (−1)(a+b−1)(x−1)(x ∗ a) ∧ (b ∗ y)− (−1)(a+b−1)(x−1)+(y−1)b((x ∗ a) ∗ y) ∧ b
− (−1)(b−1)(x−1)a ∧ ((x ∗ b) ∗ y)− (−1)(b−1)(x−1)+(y−1)(x+b−1)(a ∗ y) ∧ (x ∗ b)
− (−1)(a+b+x)(y−1)(y ∗ a) ∧ (b ∗ x)− (−1)(b+x)(y−1)a ∧ (y ∗ (b ∗ x))
− (−1)(a+b+x)(y−1)+(x−1)b(y ∗ (a ∗ x)) ∧ b− (−1)(b−1)(y−1)+(x−1)b(a ∗ x) ∧ (y ∗ b)
+ (−1)(a+b+x)(y−1)+(a+b−1)(x−1)(y ∗ (x ∗ a)) ∧ b+ (−1)(b−1)(y−1)+(a+b−1)(x−1)(x ∗ a) ∧ (y ∗ b)
+ (−1)(a+b+x)(y−1)+(b−1)(x−1)(y ∗ a) ∧ (x ∗ b) + (−1)(b+x)(y−1)+(b−1)(x−1)a ∧ (y ∗ (x ∗ b))
61
[x, [a ∧ b, y]] = x ∗ ((a ∧ b) ∗ y − (−1)(a+b−1)(y−1)y ∗ (a ∧ b))
− (−1)(a+b+y)(x−1)((a ∧ b) ∗ y − (−1)(a+b−1)(y−1)y ∗ (a ∧ b)) ∗ x
= (x ∗ a) ∧ (b ∗ y) + (−1)(x−1)(a)a ∧ (x ∗ (b ∗ y))
+ (−1)(y−1)b(x ∗ (a ∗ y)) ∧ b+ (−1)(y−1)b+(x−1)(a+y−1)(a ∗ y) ∧ (x ∗ b)
− (−1)(a+b−1)(y−1)(x ∗ (y ∗ a)) ∧ b− (−1)(a+b−1)(y−1)+(x−1)(y+a−1)(y ∗ a) ∧ (x ∗ b)
− (−1)(b−1)(y−1)(x ∗ a) ∧ (y ∗ b)− (−1)(b−1)(y−1)+(x−1)aa ∧ (x ∗ (y ∗ b))
− (−1)(a+b+y)(x−1)a ∧ ((b ∗ y) ∗ x)− (−1)(a−1)(x−1)(a ∗ x) ∧ (b ∗ y)
− (−1)(a+b+y)(x−1)+(y−1)b(a ∗ y) ∧ (b ∗ x)− (−1)(a+y)(x−1)+(y−1)b((a ∗ y) ∗ x) ∧ b
+ (−1)(a+b+y)(x−1)+(a+b−1)(y−1)(y ∗ a) ∧ (b ∗ x) + (−1)(a+y)(x−1)+(a+b−1)(y−1)((y ∗ a) ∗ x) ∧ b
+ (−1)(a+b+y)(x−1)+(b−1)(y−1)a ∧ ((y ∗ b) ∗ x) + (−1)(a−1)(x−1)+(b−1)(y−1)(a ∗ x) ∧ (y ∗ b)
We can now compute the difference [[a ∧ b, x], y] + (−1)(a+b−1)(x−1)[x, [a ∧ b, y]] − [a ∧ b, [x, y]].
Inspecting the expansions above, one easily verifies that terms of shape (� ∗ �) ∧ (� ∗ �) cancel
each other out. What’s left is killed by the Jacobi identity
[[a ∧ b, x], y] + (−1)(a+b−1)(x−1)[x, [a ∧ b, y]]− [a ∧ b, [x, y]] =
a ∧(
(b ∗ x) ∗ y − (−1)(b−1)(x−1)(x ∗ b) ∗ y − (−1)(b+x)(y−1)y ∗ (b ∗ x) + (−1)(b+x)(y−1)+(b−1)(x−1)y ∗ (x ∗ b)
+ (−1)(x−1)(b−1)x ∗ (b ∗ y)− (−1)(x+y)(b−1)x ∗ (y ∗ b)− (−1)(x−1)(y−1)(b ∗ y) ∗ x+ (−1)(y−1)(x+b)(y ∗ b) ∗ x
− b ∗ (x ∗ y) + (−1)(x−1)(y−1)b ∗ (y ∗ x) + (−1)(b−1)(x+y)(x ∗ y) ∗ b− (−1)(b−1)(x+y)+(x−1)(y−1)(y ∗ x) ∗ b)
+ (−1)(x+y)b(
(a ∗ x) ∗ y − (−1)(a−1)(x−1)(x ∗ a) ∗ y − (−1)(a+x)(y−1)y ∗ (a ∗ x)
+ (−1)(a+x)(y−1)y ∗ (x ∗ a) + (−1)(a−1)(x−1)x ∗ (a ∗ y)− (−1)(a−1)(x+y)x ∗ (y ∗ a)
− (−1)(y−1)(x−1)(a ∗ y) ∗ x+ (−1)(a+y)(x−1)+(a−1)(x+y)(y ∗ a) ∗ x− a ∗ (x ∗ y)
+ (−1)(x−1)(y−1)a ∗ (y ∗ x) + (−1)(a−1)(x+y)(x ∗ y) ∗ a− (−1)(a−1)(x+y)+(x−1)(y−1)(y ∗ x) ∗ a)∧ b
= a ∧(
[[b, x], y] + (−1)(b−1)(x−1)[x, [b, y]]− [b, [x, y]])
+(
[[a, x], y] + (−1)(a−1)(x−1)[x, [a, y]]− [a, [x, y]])∧ b
= 0
62
The compatibility of the operads Com and Lie in the Gerstenhaber case is not surprising, since
Gerstenhaber algebras come from something topological (see example 27). The compatibility of
the operads Com and LieAdm therefore begs the question: is there a topological construction—
perhaps related to the operad of little disks (little annuli, for example?)—linked to the operad
of pre-Gerstenhaber algebras? We conjecture that there is.
14 A Lie-admissable Structure on Grairredacyc
We now define a Lie-admissable structure on the symmetric collection Grairred
acyc , which general-
izes the pre-Lie product of rooted trees described by Chapoton and Livernet in [6]. We start by
considering a much smaller set of multigraphs.
Definition 39. An elementary multigraph is a multigraph contained in the graded vector space
Graacyc[1]
Elementary multigraphs look like:
•1
�
· · ·
Suppose G1 ∈ Gra[n] is irreducible and G2 is elementary. For every vertex v ∈ G1, construct
the multigraph G1 ∗v G2 ∈ Gra[n+ 1] by
1. Relabeling the singular ordinary vertex in G2 by the natural number n+ 1
2. Removing the root of G2
3. Grafting the leftmost free input in G2 onto the vertex v of G1, to the right of all existing
inputs to v
4. Grafting all remaining free inputs in G2 onto the root of G1, to the right of all existing
inputs, preserving the original ordering on grafted inputs from G2
We define the product G1 ∗G2 by the formula
G1 ∗G2 :=∑v
G1 ∗v G2 (16)
This operation extends to a product on the symmetric collection Grairredacyc . The multiplication
maps Grairredacyc [i]⊗Grairredacyc [j]→ Grairredacyc [i+ j] are defined through the following general proce-
63
dure. For multigraphs, G1 ∈ Grairredacyc [i] and G2 ∈ Grairredacyc [j], the multigraph G1∗G2 ∈ Gra[i+j]
is constructed by:
1. Relabeling the ordinary vertices 1, ..., j in G2 by i+ 1, ..., i+ j
2. Erasing all of the edges and vertices in G2 not directly connected to the root, creating a
multigraph G′2 which we regard as a wedge product of elementary rooted multigraphs
3. Applying the the left Poisson relation and formula (16) to obtain G1 ∗G′2
4. Re-inserting all of the edges and vertices erased to obtain G1 ∗G2
We illustrate this procedure with a few simple examples.
Example 28.
•1
�
· · · ∗
•1
�
· · · =
•1
�
•2
· · · · · ·
Example 29.
•1
•2
�
∗
• 1
�
· · · =
•1
•2
�
•3
· ··
+
•1
•2
�
• 3
· ··
Example 30.
•1
•2
�
∗
•1 • 2
�
=
•1
•2
�
• 4
•3
+
•1
•2
�
• 4
• 3
−
•1
•2
�
• 4
•3
−
•1
•2
�
• 4
• 3
Finally, ∗ is extended to Grairredacyc [i] ⊗ Grairredacyc [j] by bilinearity. ∗ induces a multiplication on
64
the quotient space Gra, which we will also denote by ∗, in the obvious way.
Definition 40. For every G1, G2 ∈ Gra, let G1 ∗G2 = G1 ∗G2.
It is a simple exercise to verify that G1 ∗G2 is well-defined.
Theorem 6. Grairred
acyc forms a twisted Lie-admissable algebra under the ∗-product defined above.
Proof. Let G1, G2, and G3 be rooted multigraphs in Grairred
acyc , of degree p, q, and r, respectively.
We will use the following shorthand to denote the product G1 ∗G2
1 2
This diagram actually represents a sum over all possible graftings of G2 onto G1, with signs as
specified above. Hence
(G1 ∗G2) ∗G3 =
1 2
3+
1 2
3
And
G1 ∗ (G2 ∗G3) =
1 2
3+
1 2
3
Hence,
(G1 ∗G2) ∗G3 −G1 ∗ (G2 ∗G3) =
1 2
3−
1 2
3
The Jacobi identity is the twelve term sum
[[G1, G2], G3] + (−1)(p−1)(q−1)[G2, [G1, G3]]− [G1, [G2, G3]] = 6 (∗(∗ ⊗ 1)− ∗(1⊗ ∗)) ΦGra
irredacyc
(G1 ⊗G2 ⊗G3)
=∑τ∈S3
τ
1 2
3−
1 2
3
65
with S3 acting on Grairred
acyc as described in section 9.To demonstrate Lie-admissability, it therefore
suffices to show that
1 2
3− (−1)(q−1)(r−1)
1 3
2= 0 (17)
And
1 2
3− (−1)(p−1)(q−1)
2 1
3= 0 (18)
Equation (17) is more or less trivial. Since all ordinary inputs are symmetric, the only sign
incurred by swapping G2 and G3 comes from the antisymmetry of the root vertex of G1. One
root-input from both G2 and G3 is diverted to an ordinary vertex of G1, leaving q − 1 and
r − 1 inputs, respectively, attached to the root. Swapping these inputs results in a sign of
(−1)(q−1)(r−1), from which equation (17) immediately follows.
Equation (18) is a bit more subtle. There are two cases to consider. The edges connecting
G3 to G1 and G3 to G2 may have either the same or different source vertex. Suppose, first,
that these connecting edges have different source vertices. These edges were originally inputs to
the root vertex of G3 in positions, say, a and b in the total ordering on these inputs. Without
loss of generality, suppose a < b. Thus, for any vertices v1 ∈ G1 and v2 ∈ G2 (it doesn’t
matter which ones), the term on the left of equation (18) coming from G1 ∗v1 (G2 ∗v2 G3)
has sign (−1)b−1(−1)q+a−1. The term on the right coming from G2 ∗v2 (G1 ∗v1 G3) has sign
(−1)a−1(−1)p+b. The term on the right has an additional sign of (−1)pq incurred from swapping
inputs to the root. Therefore, we have
(−1)b−1(−1)q+a−1G1 ∗v1 (G2 ∗v2 G3)− (−1)(p−1)(q−1)(−1)a−1(−1)p+b(−1)pqG1 ∗v1 (G2 ∗v2 G3)
= (−1)a+b+qG1 ∗v1 (G2 ∗v2 G3)(
1− (−1)(p−1)(q−1)+p+q+qp−1)
= (−1)a+b+qG1 ∗v1 (G2 ∗v2 G3) (1− 1)
= 0
Next, suppose the connecting edges share a source. These edges were originally inputs to the
root vertex of G3. Reflecting on the construction of G1 ∗ (G2 ∗G3), it is clear that these inputs
66
were consecutive, say a and a + 1 in the total ordering on the root-inputs in G3. The term on
the left of equation (18) coming from G1 ∗v1 (G2 ∗v2 G3) has sign (−1)a−1(−1)q+a−1. The term
on the right coming from G2 ∗v2 (G1 ∗v1 G3) has sign (−1)a−1(−1)p+a−1 and then an additional
sign of (−1)pq+1 incurred from swapping inputs to the root and swapping the connecting edges
from G3. Therefore, we have
(−1)a−1(−1)q+a−1G1 ∗v1 (G2 ∗v2 G3)− (−1)(p−1)(q−1)(−1)a−1(−1)p+a−1(−1)pq+1G1 ∗v1 (G2 ∗v2 G3)
= (−1)qG1 ∗v1 (G2 ∗v2 G3)(
1− (−1)(p−1)(q−1)+p+q+qp−1)
= (−1)a+b+qg1 ∗v1 (G2 ∗v2 G3) (1− 1)
= 0
which completes the proof.
Corollary 5. The wedge product and the ∗-product turn the symmetric collection Graacyc into
a twisted pre-Gerstenhaber algebra.
Proof. This follows immediately from theorem 5 and theorem 6.
Notice that:
1. The restriction of ∗ to the space of rooted trees (i.e., the subcollection of Grairred
acyc spanned
in each component by rooted multigraphs containing no vertex with more than one outgoing
edge) is precisely the pre-Lie concatenation product defined by Chapoton and Livernet in
[6].
2. The Lie algebra product [a, b] = a ∗ b− (−1)(|a|−1)(|b|−1)b ∗ a corresponds to the usual Lie
bracket of differential operators.
15 A Co-algebra Structure on Grairredacyc
Let V = {Vn}n≥0 be a symmetric collection in the category of graded vector spaces. The dual
of V , denoted V ∗, is defined by (V ∗)n = (Vn)∗, where (Vn)∗ is the dual graded vector space
associated to Vn. Duality defines a functor SymCol→ SymColop with f∗ : W ∗ → V ∗ defined
by (f∗)n = (fn)∗ for every morphism f : V →W . Recall the following notation: for every n ∈ Nand k ∈ Z, the subspace of Vn consisting of homogenous degree-k elements is denoted by V kn .
67
Definition 41. A symmetric collection V is locally finite-dimensional if for every n ∈ N and
k ∈ Z, V kn is a finite-dimensional vector space.
Proposition 18. If V is locally-finite dimensional, the natural map ϕ : V → V ∗∗ is an isomor-
phism of symmetric collections.
Proof. For every n ∈ N, ϕn : Vn → (V ∗∗)n = (Vn)∗∗ is a map of graded vector spaces ϕn =
⊕k∈Zϕkn, where ϕkn : V kn → V k∗∗n is given by the evaluation:
ϕkn : vk 7−→(wk 7→ wk(vk)
)for every vk ∈ V kn and wk ∈ V k∗n . Each ϕkn is injective by the standard argument involving
the axiom of choice, and therefore surjective by the finiteness of dimV kn and the rank-nullity
theorem
Thus, duality induces an equivalence of categories SymCollocally finite∼= SymColoplocally finite.
Duality behaves nicely with respect to tensor products. For every pair of symmetric collections
V,W ∈ SymCol, there is a natural morphism
ψ : V ∗ ⊗W ∗ → (V ⊗W )∗
which satisfies
ψn(v∗ ⊗ w∗)(v ⊗ w) = v∗(v)w∗(w)
for v ∈ Vi, v∗ ∈ V ∗i , w ∈Wj , and w∗ ∈W ∗j and i+ j = n.
Proposition 19. For V,W locally finite-dimensional, the natural morphism ψ : V ∗ ⊗W ∗ →(V ⊗W )∗ is an isomorphism of symmetric collections.
Proof. Recall,
(V ∗ ⊗W ∗)n =⊕i+j=n
IndSnSi×SjV∗i ⊗W ∗j
and
68
(V ⊗W )∗n =
⊕i+j=n
IndSnSi×SjVi ⊗Wj
∗ =⊕i+j=n
IndSnSi×Sj (Vi ⊗Wj)∗
Each V ∗i ⊗W ∗j , being a tensor product of graded vector spaces, is a direct sum
Vi ⊗Wj =⊕
k1+k2=k
V k1∗i ⊗W k2∗j (19)
And each (Vi ⊗Wj)∗ is given by
(Vi ⊗Wj)∗ =
( ⊕k1+k2=k
V k1i ⊗Wk2j
)∗=
⊕k1+k2=k
(V k1i ⊗Wk2j )∗ (20)
Moreover, since the direct sums in decompositions (19) and (20) are direct sums of Si × Sj-
representations, these sums commute with induction, i.e.
(V ∗ ⊗W ∗)n =⊕i+j=nk1+k2=k
IndSnSi×SjVk1∗i ⊗W k2∗
j =⊕i+j=nk1+k2=k
⊕(i,j)−shuffles σ
σ V k1∗i ⊗W k2∗j
and
(V ⊗W )∗n =⊕i+j=nk1+k2=k
IndSnSi×Sj (Vk1i ⊗W
k2j )∗ =
⊕i+j=nk1+k2=k
⊕(i,j)−shuffles σ
σ (V k1i ⊗Wk2j )∗
The nth component of the canonical map ψ : V ∗ ⊗W ∗ → (V ⊗W )∗ is a direct sum
ψn =⊕i+j=nk1+k2=k
⊕(i,j)−shuffles σ
σ ψk1,k2i,j
where σ ψk1,k2i,j : σ V k1∗i ⊗W k2∗j → σ (V k1i ⊗W
k2j )∗ is the linear map given by
σ ψk1,k2i,j (σ vk1∗i ⊗ wk2∗j )(σ vk1i ⊗ wk2j ) = vk1∗i (vk1i )wk2∗j (wk2j )
for every vk1i ∈ Vk1i , vk1∗i ∈ V k1∗i , wk2j ∈ W
k2j , and wk2∗j ∈ W k2∗
j . The important point is that
each σ ψk1,k2i,j is a linear map between finite-dimensional vector spaces and hence an isomorphism
by the standard argument involving the axiom of choice.
69
Recall from section 9 the construction of the epimorphism ΦV : ⊗3V → A3(V ): for every
symmetric collection V , there is an action of S3 on ⊗3V given on simple transpositions τ(l) = l+1
and elements σ v1 ⊗ v2 ⊗ v3 ∈ IndSi+j+kSi×Sj×Sk (Vi ⊗ Vj ⊗ Vk) ⊂ ⊗3V by
τ (σ v1 ⊗ v2 ⊗ v3) = −(−1)(|vl|−1)(|vl+1|−1)τi,j,kσ vτ(1) ⊗ vτ(2) ⊗ vτ(3)
and extended uniquely to S3. The fixed points form a subcollection A3(V ) ⊂ ⊗3V and ΦV :
⊗3V → A3(V ) is the morphism defined by
ΦV (σ v1 ⊗ v2 ⊗ v3) =∑τ∈S3
1
6τ(σ v1 ⊗ v2 ⊗ v3)
We have the following proposition:
Proposition 20. Suppose V and W are locally finite-dimensional and f : V → W an isomor-
phism. Then Φ satisfies
1. ΦV = ⊗3f−1 ◦ ΦW ◦ ⊗3f (naturality)
2. ΦV ∗ = (ΦV )∗ (self-duality)
Proof. It is easy to see that the isomorphism ⊗3f commutes with the action of S3. Indeed, for
any σ v1 ⊗ v2 ⊗ v3 ∈ ⊗3V , and simple transposition τ ∈ S3, τ(l) = l + 1 we have:
⊗3f (τ (σ v1 ⊗ v2 ⊗ v3)) = ⊗3f(−(−1)(|vl|−1)(|vl+1|−1)τi,j,kσ vτ(1) ⊗ vτ(2) ⊗ vσ(3)
)= −(−1)(|vl|−1)(|vl+1|−1)τi,j,kσ ⊗3 f
(vτ(1) ⊗ vτ(2) ⊗ vτ(3)
)= −(−1)(|vi|−1)(|vi+1|−1)τi,j,kσ f(vσ(1))⊗ f(vσ(2))⊗ f(vσ(3))
= τ(⊗3f(σ v1 ⊗ v2 ⊗ v3)
)And, therefore,
70
⊗3f−1 ◦ ΦW ◦ ⊗3f(σ v1 ⊗ v2 ⊗ v3) = ⊗3f−1 ◦ ΦW (σ f(v1)⊗ f(v2)⊗ f(v3))
= ⊗3f−1
(∑τ∈S3
1
6τ (σ f(v1)⊗ f(v2)⊗ f(v3))
)
=∑τ∈S3
1
6⊗3 f−1 (τ (σ f(v1)⊗ f(v2)⊗ f(v3)))
=∑τ∈S3
1
6τ (⊗3f−1(σ f(v1)⊗ f(v2)⊗ f(v3)))
=∑τ∈S3
1
6τ (σ v1 ⊗ v2 ⊗ v3)
= ΦV (σ v1 ⊗ v2 ⊗ v3)
which establishes 1. Now, select a basis {vi}i∈N for V (i.e. select a basis for each Vn and take
the disjoint union). For each vi, let vi denote the associated covector vi(vj) = δ(i, j). Since V
is locally finite-dimensional, {vi}i∈N forms a dual basis for V ∗ and {σ vi1 ⊗ vi2 ⊗ vi3} forms a
basis for ⊗3V ∗. By definition
(ΦV )∗(σ vi1 ⊗ vi2 ⊗ vi3)(σ′ vj1 ⊗ vj2 ⊗ vj3) = (σ vi1 ⊗ vi2 ⊗ vi3) (ΦV (σ′ vj1 ⊗ vj2 ⊗ vj3))
This is non-zero if and only if σ = σ′ and ∃τ ∈ S3 such that ik = jτ(k) for k = 1, 2, 3. Assume
that τ is a simple transposition τ(t) = t+ 1 (the general case follows since the symmetric group
is generated by simple transpositions). Hence,
(σ vi1 ⊗ vi2 ⊗ vi3) ((ΦV )∗(σ vj1 ⊗ vj2 ⊗ vj3)) = −(−1)(|vjt |−1)(|vjt+1|−1)
= −(−1)(|vjτ(t+1)
|−1)(|vjτ(t) |−1)
= −(−1)(|vit+1 |−1)(|vit |−1)
= ΦV ∗(σ vi1 ⊗ vi2 ⊗ vi3)(σ vj1 ⊗ vj2 ⊗ vj3)
which establishes 2.
Recall from section 9, a twisted algebra (A,×) is graded Lie-admissable if, as a map ⊗3A →A,
71
(×(×⊗ 1)−×(1⊗×)) ΦA = 0
This motivates the following obvious definition:
Definition 42. A twisted co-algebra (A, ∂) is co-lie-admissable if, as a map A→ ⊗3A,
ΦA ((∂ ⊗ 1)∂ − (1⊗ ∂)∂) = 0
The ∗-product on rooted multigraphs defines a morphism
∗ : Grairred
acyc ⊗Grairred
acyc → Grairred
acyc
Since Grairred
acyc is locally finite-dimensional, dualizing yields a morphism
∗∗ : Grairred∗acyc → Gra
irred∗acyc ⊗Gra
irred∗acyc
Let B denote the standard basis of multigraphs for Grairredacyc . Applying the projection map
π : Grairredacyc → Grairred
acyc to B yields a spanning set for Grairred
acyc . We can obtain a basis
B for Grairred
acyc by selecting an element from each equivalence class in the quotient π(B)/±.
Since Grairred
acyc is locally finite-dimensional, a basis B for Grairred
acyc induces an isomorphism
ψB : Grairred
acyc∼→ Gra
irred∗acyc .
Proposition 21. The isomorphism ψB : Grairred
acyc∼→ Gra
irred∗acyc is well-defined.
Proof. Let B and B′ be two bases obtained from the standard basis B. Every element x ∈ B has
an associated covector x∗ ∈ Grairredacyc∗ defined by
x∗(b) =
{1 if b = x
0 otherwise
for every b ∈ B and extended by linearity. Since Grairred
acyc is locally finite-dimensional, the set
B∗ := {x∗ | x ∈ B} forms a dual basis for Grairred∗acyc . The assignment x 7→ x∗ is therefore an
isomorphism ψB : Grairred
acyc → Grairred∗acyc which corresponds to the identity matrix with respect
to the bases B and B∗. The important fact in this case is that the change of basis matrix M
72
transforming B into B′ is a diagonal matrix with 1’s and −1’s in the diagonal. In particular,
M is orthogonal and satisfies (M−1)T = MT = M . The change of basis x′ = Mx in Grairred
acyc
corresponds to a change of basis x∗′
= (M−1)Tx∗ in the dual space Grairred∗acyc . Thus, the
isomorphism ψB′ : Grairred
acyc → Grairred∗acyc induced by B′ is represented by the matrix
(M−1)TM−1 = MM−1 = 1
It follows that the isomorphism ψB : Grairred
acyc → Grairred∗acyc is independent of our particular choice
of B.
We can apply the isomorphism ψB betweenGrairred
acyc and its dual to the morphism ∗∗ : Grairred∗acyc →
Grairred∗acyc ⊗Gra
irred∗acyc to obtain a coproduct on Gra
irred
acyc .
Definition 43. The co-product
∆ : Grairred
acyc → Grairred
acyc ⊗Grairred
acyc
is defined as the composition
∆ = ψ−1B ⊗ ψ
−1B ◦ ∗
∗ ◦ ψB
Proposition 22. The twisted co-algebra (Grairred
acyc ,∆) is co-Lie-admissable.
Proof. For notational convenience, put G = Grairred
acyc . In Theorem 6 we proved that the algebra
(G, ∗) is graded Lie-admissable. Hence, by definition,
(∗(∗ ⊗ 1)− ∗(1⊗ ∗)) ΦG = 0
Dualizing, we get
(ΦG)∗ ((∗∗ ⊗ 1) ∗∗ − (1⊗ ∗∗)∗∗) = 0
Thus,
73
0 = ⊗3ψ−1B ◦ (ΦG)∗ ((∗∗ ⊗ 1) ∗∗ − (1⊗ ∗∗)∗∗) ◦ ψB
= (⊗3ψ−1B ◦ (ΦG)∗ ◦ ⊗3ψB)
((⊗2ψ−1
B ◦ ∗∗ ◦ ψB)⊗ 1)(⊗2ψ−1
B ◦ ∗∗ ◦ ψB)− (1⊗ (⊗2ψ−1
B ◦ ∗∗ ◦ ψB))(⊗2ψ−1
B ◦ ∗∗ ◦ ψB)
)= (⊗3ψ−1
B ◦ (ΦG)∗ ◦ ⊗3ψB) ((∆⊗ 1)∆− (1⊗∆)∆)
However, by parts 1 and 2 of Proposition 20, we have
⊗3ψ−1B ◦ (ΦG)∗ ◦ ⊗3ψB = ΦG
So,
ΦG ((∆⊗ 1)∆− (1⊗∆)∆)
as desired.
For a particular choice of B, ∆ is given on basis elements x ∈ Grairredacyc [n] by the formula
∆(x) :=∑a,b∈B
ε(a,b)x∈σa∗b
σ ε(a, b)a⊗b ∈⊕i+j=n
IndSnSi×SjGrairred
acyc [i]⊗Grairredacyc [j] =(Gra
irred
acyc ⊗Grairred
acyc
)n
(21)
where ε(a, b) is ±1, σ ∈ Sn is a permutation shuffling the vertices of a and b, and the notation
ε(a, b)x ∈ σa ∗ b indicates that the term ε(a, b)x is represented in the unique B-decomposition of
σ a ∗ b.
Intuitively, ∆ is the opposite of ∗: whereas G1 ∗G2 is the sum of all possible graftings of G2 onto
G1, ∆(G) is the sum of all possible ‘splitting aparts’ of G.
Example 31.
∆
•2
•1
� =
•1
�⊗
•1
�
74
Example 32.
∆
•2 • 3
• 1
� =
•2
•1
�
⊗•1
�+ (23)
•2
•1
�
⊗•1
�
with ‘(23)’ denoting, in cycle notation, the (2, 1)-shuffle σ ∈ S3 given by 1 7→ 1, 2 7→ 3, 3 7→ 2.
Example 33.
∆
•2
•3
�
• 1
•4
= (234) •2
�
• 1
•3
⊗•1
�− (123)
•1
•2
�
⊗
•2 • 1
�
+
•2
•3
�
• 1 ⊗•1
�
16 Loday’s Rigidity Theorem
In [7], Loday formalizes the notion of a generalized bialgebra, generalizing the classical notion of
a Hopf algebra, among other things. A type of generalized bialgebra is determined by an algebra
structure, a co-algebra structure, and a compatibility relation between them. The algebra and
co-algebra structures are encoded in the data of a finitely generated operad A and finitely gen-
erated co-operad Cc (see [7] for relevant definitions). For example, if A = As and Cc = Asc with
the standard Hopf compatibility relation, we recover the familiar Hopf algebra. Loday’s Rigidity
Theorem, a special case of the much more general Structure Theorem for generalized bialgebras,
is a powerful tool for deducing the freeness of certain types of bialgebras. We assume all operads
and co-operads are graded vector spaces over R.
Definition 44. A compatibility relation is an equation involving compositions of operations and
co-operations. A compatibility relation between an operation µ and co-operation δ is distributive
if it has the form
75
δ ◦ µ =∑i
(µi1 ⊗ ...⊗ µim) ◦ ωi ◦ (δi1 ⊗ ...⊗ δin)
where
µ ∈ A(n), µij ∈ A(kj),
δ ∈ C(m), δij ∈ C(lj),m∑i=1
ki = r =
n∑i=1
li,
ωi ∈ R[Sr]
Definition 45. Let H be a generalized A− Cc bialgebra. The primitive part of H, denoted by
Prim(H), is the subset
PrimH := {x ∈ H | δ(x) = 0 ∀δ ∈ Cc(n), n ≥ 2}
Rigidity Theorem for Generalized Bialgebras. LetA−Cc be a type of generalized bialgebra
such that the following hypotheses hold:
H0 The operad C is finitely generated and for every generating operation µ and generating
co-operation δ, there is a distributive compatibility relation.
H1 The free A-algebra A(V ) is naturally endowed with an A−Cc bialgebra structure, inducing
a natural co-algebra map ϕ : A(V )→ Cc(V ).
H2 The natural co-algebra map ϕ : A(V )→ Cc(V ) is an isomorphism.
Then any A− Cc bialgebra H is both free and co-free over its primitive elements, i.e.
A(PrimH) ∼= H ∼= Cc(PrimH)
Note that Loday’s Rigidity Theorem generalizes trivially to twisted bialgebras.
Grairred
acyc contains two important subalgebras.
Definition 46. A1 ⊆ Grairred
acyc is the subcollection spanned by irreducible, internally acyclic
multigraphs containing no vertices with more than one outgoing edge.
76
Definition 47. A≥2 ⊂ Grairred
acyc is the subcollection spanned by irreducible, internally acyclic
multigraphs containing no ordinary vertices with fewer than two outgoing edge.
It is clear from definitions that A1 and A≥2 are both subalgebras and subcoalgebras of Grairred
acyc .
Definition 48. For every pair of multigraphs a, b ∈ Grairred
acyc , let n(a, b) denote the product
(#V ert(a)) (#Out(b)) where, as usual, V ert(a) denotes the set of vertices in a and Out(b)
denotes the set of vertices in b directly connected to the root.
Note that if b ∈ A1, n(a, b) = #V ert(a).
Theorem 7. For every a, b ∈ A1, ∆ and ∗ satisfy the following compatibility relation:
∆(a ∗ b) = ∆(a) ∗ (1⊗ b) + ∆(a) ∗ (b⊗ 1) + (a⊗ 1) ∗∆(b) + n(a, b)a⊗ b (22)
Proof. Since ∆, ∗, and ⊗ are bilinear, it suffices to prove equation (22) on a basis. Let B be any
basis for Grairred
acyc obtained from the standard basis of multigraphs for Grairredacyc and let a, b ∈ B.
By definition
∆(a ∗ b) = ∆
(∑v∈a
a ∗v b
)=∑v∈a
∆(a ∗v b)
We have ∆(a) =∑i ai(1) ⊗ a
i(2) where ai(1) and ai(2) are full subgraphs of a (no signs are present
since a ∈ A1). In Sweedler’s notation, all indices are suppressed: ∆(a) =∑
(a) a(1)⊗a(2). Hence,
∆(a∗v b) =∑
(a∗vb)(a∗v b)(1)⊗ (a∗v b)(2). The edge connecting b to the vertex v in a is contained
in (a ∗v b)(1), (a ∗v b)(2), or neither. Thus,
∆(a ∗v b) =∑a(2)3v
a(1) ⊗ (a(2) ∗v b) +∑a(1)3v
(a(1) ∗v b)⊗ a(2) +∑(b)
(a ∗v b(1))⊗ b(2) + a⊗ b
Hence,
77
∆(a ∗ b) =∑v∈a
∑a(2)3v
a(1) ⊗ (a(2) ∗v b) +∑a(1)3v
(a(1) ∗v b)⊗ a(2) +∑(b)
(a ∗v b(1))⊗ b(2) + a⊗ b
=∑(a)
∑v∈a(2)
a(1) ⊗ (a(2) ∗v b) +∑(a)
∑v∈a(1)
(a(1) ∗v b)⊗ a(2) +∑(b)
∑v∈a
(a ∗v b(1))⊗ b(2) +∑v∈a
a⊗ b
=∑(a)
a(1) ⊗ (a(2) ∗ b) +∑(a)
(a(1) ∗ b)⊗ a(2) +∑(b)
(a ∗ b(1))⊗ b(2) + n(a, b)a⊗ b
= ∆(a) ∗ (1⊗ b) + ∆(a) ∗ (b⊗ 1) + (a⊗ 1) ∗∆(b) + n(a, b)a⊗ b
Theorem 8. Bialgebras of type PreLie–PreLiec with compatibility relation defined by equation
(22) satisfy the hypotheses of Loday’s Rigidity Theorem.
Sketch of Proof. Clearly, equation (22) js distributive. We must exhibit a natural co-pre-Lie
structure on the free pre-Lie algebra PreLie(V ) over a symmetric collection V . Let I ⊂PreLie(V ) denote the ideal spanned by elements of form (∗(∗ ⊗ 1)− ∗(1⊗ ∗)) (a⊗b⊗c−a⊗c⊗b).The assignment ∆(v) = 0,∀v ∈ V extends to a coproduct on PreLie(V ) if and only if ∆(I) = 0.
And indeed, for every a, b, c ∈ PreLie(V ),
∆((ab)c− a(bc)) = ∆(ab)(1⊗ c) + ∆(ab)(c⊗ 1) + (ab⊗ 1)∆(c) + n(ab)ab⊗ c
− (∆(a)(1⊗ bc) + ∆(a)(bc⊗ 1) + (a⊗ 1)∆(bc) + n(a)a⊗ bc)
= ∆(a) (1⊗ bc+ b⊗ c+ c⊗ b+ bc⊗ 1) + (a⊗ 1)∆(b)(1⊗ c+ c⊗ 1)
+ n(a)(a⊗ bc) + n(a)(ac⊗ b) + (ab⊗ 1)∆(c) + n(ab)(ab⊗ c)
− (∆(a)(1⊗ bc+ bc⊗ 1) + (a⊗ 1)∆(b)(1⊗ c+ c⊗ 1)
+ (ab⊗ 1)∆(c) + n(b)(ab⊗ c) + n(a)(a⊗ bc))
= ∆(a)(b⊗ c+ c⊗ b) + n(a)(ac⊗ b) + n(ab)(ab⊗ c)− n(b)(ab⊗ c)
where, for notational convenience, all products have been written as concatenations and n(a) =
#V ert(a). Since n(ab) = n(a) + n(b), the last three terms simplify to
n(a)(ac⊗ b) + n(ab)(ab⊗ c)− n(b)(ab⊗ c) = n(a)(ac⊗ b+ ab⊗ c)
Thus,
78
∆((ab)c− a(bc)) = ∆(a)(b⊗ c+ c⊗ b) + n(a)(ac⊗ b+ ab⊗ c)
which is symmetric in the second two factors. So, indeed, ∆(v) = 0 extends to a coproduct on
PreLie(V ). We must show that this coproduct is co-pre-Lie. We proceed by induction, noting
that the base case is trivial, since (∆⊗ 1)∆(v)− (1⊗∆)∆(v) = 0 for every v ∈ V . Suppose that
the co-associators of a and b are co-pre-Lie and consider the co-associator of their product ab.
We compute, in Sweedler’s notation,
(∆⊗ 1)∆(ab)− (1⊗∆)∆(ab) = (∆⊗ 1− 1⊗∆)(∆(a)(1⊗ b+ b⊗ 1) + (a⊗ 1)∆(b) + n(a)a⊗ b)
= (∆⊗ 1− 1⊗∆)(a(1) ⊗ a(2)b+ a(1)b⊗ a(2) + ab(1) ⊗ b(2) + n(a)a⊗ b)
= ∆(a(1))⊗ a(2)b+ ∆(a(1)b)⊗ a(2) + ∆(ab(1))⊗ b(2) + n(a)∆(a)⊗ b
−(a(1) ⊗∆(a(2)b) + a(1)b⊗∆(a(2)) + ab(1) ⊗∆(b(2)) + n(a)a⊗∆(b)
)= ∆(a(1))⊗ a(2)b+ ∆(a(1))(1⊗ b+ b⊗ 1)⊗ a(2) + a(1)b(1) ⊗ b(2) ⊗ a(2)
+ n(a(1))a(1) ⊗ b⊗ a(2) + a(1) ⊗ a(2)b(1) ⊗ b(2) + a(1)b(1) ⊗ a(2) ⊗ b(2)
+ (a⊗ 1)∆(b(1))⊗ b(2) + n(a)a⊗ b(1) ⊗ b(2) + n(a)a(1) ⊗ a(2) ⊗ b
−(a(1) ⊗∆(a(2))(1⊗ b+ b⊗ 1) + a(1) ⊗ a(2)b(1) ⊗ b(2) + n(a(2))a(1) ⊗ a(2) ⊗ b
+a(1)b⊗∆(a(2)) + ab(1) ⊗∆(b(2)) + n(a)a⊗ b(1) ⊗ b(2)
)= ∆(a(1))⊗ a(2)b+ ∆(a(1))(1⊗ b+ b⊗ 1)⊗ a(2) + a(1)b(1) ⊗ b(2) ⊗ a(2)
+ n(a(1))a(1) ⊗ b⊗ a(2) + a(1)b(1) ⊗ a(2) ⊗ b(2)
+ (a⊗ 1)∆(b(1))⊗ b(2) + n(a)a(1) ⊗ a(2) ⊗ b
−(a(1) ⊗∆(a(2))(1⊗ b+ b⊗ 1) + n(a(2))a(1) ⊗ a(2) ⊗ b
+a(1)b⊗∆(a(2)) + ab(1) ⊗∆(b(2)))
The three terms with non-trivial coefficients simplify, via the distributivity of n(a):
n(a(1))a(1)⊗b⊗a(2)+n(a)a(1)⊗a(2)⊗b−n(a(2))a(1)⊗a(2)⊗b = n(a(1))(a(1) ⊗ a(2) ⊗ b+ a(1) ⊗ b⊗ a(2)
)which is symmetric in the second two factors. This leaves
79
∆(a(1))⊗ a(2)b+ ∆(a(1))(1⊗ b+ b⊗ 1)⊗ a(2) + a(1)b(1) ⊗ b(2) ⊗ a(2)
+ a(1)b(1) ⊗ a(2) ⊗ b(2) + (a⊗ 1)∆(b(1))⊗ b(2)
−(a(1) ⊗∆(a(2))(1⊗ b+ b⊗ 1) + a(1)b⊗∆(a(2)) + ab(1) ⊗∆(b(2))
)= (a⊗ 1⊗ 1)
(∆(b(1))⊗ b(2) − b(1) ⊗∆(b(2))
)+ a(1)b(1) ⊗ b(2) ⊗ a(2) + a(1)b(1) ⊗ a(2) ⊗ b(2)
+ ∆(a(1))⊗ a(2)b+ ∆(a(1))(1⊗ b+ b⊗ 1)⊗ a(2)
−(a(1) ⊗∆(a(2))(1⊗ b+ b⊗ 1) + a(1)b⊗∆(a(2))
)The first term is co-pre-Lie by induction and the following two terms (considered as a sum)
a(1)b(1) ⊗ b(2) ⊗ a(2) + a(1)b(1) ⊗ a(2) ⊗ b(2) are clearly symmetric in the second two factors. We
are left with the expression
∆(a(1))⊗ a(2)b+ ∆(a(1))(1⊗ b+ b⊗ 1)⊗ a(2) − a(1) ⊗∆(a(2))(1⊗ b+ b⊗ 1)− a(1)b⊗∆(a(2))
This factors as(∆(a(1))⊗ a(2) − a(1) ⊗∆(a(2))
)(b⊗ 1⊗ 1 + 1⊗ b⊗ 1 + 1⊗ 1⊗ b) which is, again,
co-pre-Lie by induction.
Thus, PreLie(V ) is canonically endowed with the structure of a twisted co-pre-Lie co-algebra.
To complete the proof, we must show that the canonical surjection p : PreLie(V )→ PreLiec(V )
is an isomorphism. For every operation µ ∈ PreLie(n) and co-operation β ∈ PreLiec(n), we
have
(β ◦ µ)(x1 ⊗ ...⊗ xn) =∑σ∈Sn
cσxσ(1) ⊗ ...⊗ xσ(n) (23)
The assignment 〈µ, β〉 = cid defines a pairing 〈 , 〉 : PreLie(n) ⊗ PreLiec(n) → R. Suppose
0 6= u = µ(x1 ⊗ ...⊗ xn) and p(u) = 0. Take any non-trivial co-operation β of the same arity as
µ. Since p is a morphism of co-algebras,
(p⊗ ...⊗ p)((β ◦ µ)(x1 ⊗ ...⊗ xn)) = β(p(µ(x1 ⊗ ...⊗ xn))) = β(p(u)) = 0
Hence, by equation (23),
∑σ∈Sn
cσxσ(1) ⊗ ...⊗ xσ(n) = (p⊗ ...⊗ p)∑σ∈Sn
cσxσ(1) ⊗ ...⊗ xσ(n) = 0
80
And in particular, 〈µ, β〉 = 0. Thus, it suffices to show that this pairing is non-degenerate.
This can be done, for example, by selecting bases for PreLie(n) and PreLiec(n), computing the
pairing on these basis elements, and then calculating the determinant. We have not yet been
able to produce a succinct formula for this pairing, so we leave this step incomplete.
Corollary 6. A1 is a freely generated as a twisted pre-Lie algebra.
Proof. This follows immediately from theorems 7, 8, and Loday’s Rigidity Theorem.
On A≥2, the relationship between ∗ and ∆ is a bit more subtle. To describe this relationship,
we must introduce a bit of additional notation. Fix a basis B of multigraphs for Grairred
acyc . For
multigraphs a, b ∈ B,
a ∗ b =∑v∈a
w∈Out(b)
sb(w) a ∗v,w b
where sb(w) = ±1 and a∗v,w b is the multigraph formed by decapitating b, grafting the left-most
free output from w onto v, and grafting all other free outputs onto the root vertex of a. The
sign sb(w) can be computed explicitly by counting the number x of free outputs to the left of
the left-most free output from w and then computing (−1)x. Thus,
∆(a ∗ b) = ∆
∑v∈a
w∈Out(b)
sb(w) a ∗v,w b
=∑v∈a
w∈Out(b)
sb(w)∆(a ∗v,w b) (24)
On the other hand,
∆(a) =∑(a)
εa(a(1), a(2))a(1) ⊗ a(2)
where a(1) and a(2) are full subgraphs of a such that, in its unique B-decomposition, εa(a(1), a(2))a(1)∗a(2) contains the term a with coefficient 1. For notational convenience, we will suppress the sub-
script a in εa(a(1), a(2)) when there is no potential for confusion.
Theorem 9. For every pair of multigraphs a, b ∈ A≥2 with the property that every ordinary
vertex is directly connected to the root, ∆ and ∗ satisfy the following compatibility relation:
81
∆(a ∗ b) =∑(a)
ε(a(1), a(2))(a(1) ⊗ (a(2) ∗ b) + (−1)(|b|−1)(|a(2)|−1)(a(1) ∗ b)⊗ a(2)
)+∑(b)
ε(b(1), b(2))(
(a ∗ b(1))⊗ b(2) + (−1)(|a|−1)(|b(1)|−1)b(1) ⊗ (a ∗ b(2)))
+ n(a, b)a⊗ b
(25)
In other words, the five-term sum ∆(a ∗ b) = ∆(a) ∗ (1 ⊗ b) + ∆(a) ∗ (b ⊗ 1) + (a ⊗ 1) ∗∆(b) +
(1⊗ a) ∗∆(b) + n(a, b)a⊗ b with the appropriate Koszul signs.
Proof. The multigraph a∗v,wb can be broken up into two pieces in five different ways, as expressed
in the following sum
∆(a ∗v,w b) =∑a(2)3v
ε(a(1), (a(2) ∗v,w b))a(1) ⊗ (a(2) ∗v,w b) +∑a(1)3v
ε((a(1) ∗v,w b), a(2))(a(1) ∗v,w b)⊗ a(2)
+∑b(1)3w
ε((a ∗v,w b(1)), b(2))(a ∗v,w b(1))⊗ b(2) +∑b(2)3w
ε(b(1), (a ∗v,w b(2)))b(1) ⊗ (a ∗v,w b(2))
+ ε(a, b)a⊗ b(26)
with an implied subscript of a ∗v,w b for each ε above. Unraveling definitions, we see that
ε(a(1), (a(2) ∗v,w b)) = εa(a(1), a(2))
ε((a(1) ∗v,w b), a(2)) = (−1)(|b|−1)(|a(2)|−1)εa(a(1), a(2))
ε((a ∗v,w b(1)), b(2)) = εb(b(1), b(2))
ε(b(1), (a ∗v,w b(2))) = (−1)(−1)|a|(−1)|a||b(1)|εb(b(1), b(2))
Substituting these identities into equation (26) and equation (26) into equation (24), we obtain
by commuting sums
82
∆(a ∗ b) =∑v∈a
w∈Out(b)
s(w)∆(a ∗v,w b)
=∑(a)
ε(a(1), a(2))∑v∈a(2)
w∈Out(b)
sb(w)a(1) ⊗ (a(2) ∗v,w b)
+∑(a)
(−1)(|b|−1)(|a(2)|−1)ε(a(1), a(2))∑v∈a(1)
w∈Out(b)
sb(w)(a(1) ∗v,w b)⊗ a(2)
+∑(b)
εb(b(1), b(2))∑v∈a
w∈Out(b)∩b(1)
sb(w)(a ∗v,w b(1))⊗ b(2)
+∑(b)
(−1)|a||b(1)|+|a|+1ε(b(1), b(2))∑v∈a
w∈Out(b)∩b(2)
sb(w)b(1) ⊗ (a ∗v,w b(2))
+∑v∈a
w∈Out(b)
sb(w)εa∗v,wb(a, b)a⊗ b
Applying the fact that Out(b)∩b(1) = Out(b(1)) and Out(b)∩b(2) = Out(b(2)), this can be further
simplified:
∆(a ∗ b) =∑(a)
ε(a(1), a(2))a(1) ⊗ (a(2) ∗ b) +∑(a)
ε(a(1), a(2))(−1)(|b|−1)(|a(2)|−1)(a(1) ∗ b)⊗ a(2)
+∑(b)
ε(b(1), a(2))(a ∗ b(1))⊗ b(2) + ε(b(1), a(2))(−1)(|a|−1)(|b(1)|−1)b(1) ⊗ (a ∗ b(2))
+∑v∈a
w∈Out(b)
sb(w)εa∗v,wb(a, b)a⊗ b
Since εa∗v,wb(a, b) = sb(w), the final sum reduces to n(a, b)a⊗ b, which completes the proof.
We conjecture that A≥2 is a free twisted Lie-admissable algebra, generated by elementary multi-
graphs of valence ≥ 2.
17 A Useful Consequence of Kunneth’s Theorem
We pause here to prove a simple consequence of Kunneth’s theorem for cohomology. This corol-
lary will prove useful in section 18.
83
Suppose V and W are cochain complexes of right and left R-modules, respectively. The tensor
product V ⊗RW is the cochain complex
(V ⊗RW )n =⊕i+j=n
V i ⊗RW j
with coboundary ∂ : (V ⊗R W )n → (V ⊗R W )n+1 given on pure tensors v ⊗ w ∈ V i ⊗R W j by
the formula:
∂(v ⊗ w) = (∂(v)⊗ w) + (−1)i(v ⊗ ∂(w)) ∈ (V ⊗RW )n+1
If R is commutative, V is a cochain complex of R-modules. In this case, tensor products such
as T kV can be formed by tensoring V with itself repeatedly. The tensor algebra T (V ) is the
pointwise sum of all T kV . For each k, there is an action of the symmetric group Sk on the
components of T kV , which commutes with the differential. We can therefore consider the sub-
complex SkV ⊂ T kV of symmetric invariants. The pointwise sum S(V ) :=⊕
k Sk(V ) is the
symmetric algebra of V .
Note that there is a graded version of this construction. Suppose V is a cochain complex of graded
R-modules. Then the tensor product T kV is also a complex of graded modules, with gradings
inherited from V . Sk then acts on the complex T kV via the Koszul sign rule. Explicitly, for
every simple transposition σα : α↔ α+ 1, we put
σα(v1 ⊗ v2 ⊗ ...⊗ vk) = (−1)|vα||vα+1|v1 ⊗ v2 ⊗ ...vα+1 ⊗ vα ⊗ ...⊗ vk
It is an easy verification that the formula above respects all of the relations among simple
transpositions. Indeed, ∀α < k:
σα ◦ σα(v1 ⊗ v2 ⊗ ...⊗ vk) = (−1)2|vα||vα+1|v1 ⊗ v2 ⊗ ...⊗ vk = v1 ⊗ v2 ⊗ ...⊗ vk
∀α < β < k, β 6= α+ 1:
σα ◦ σβ(v1 ⊗ v2 ⊗ ...⊗ vk) = (−1)|vα||vα+1|+|vβ ||vβ+1|v1 ⊗ ...vα+1 ⊗ vα ⊗ ...vβ+1 ⊗ vβ ⊗ ...vk
= σβ ◦ σα(v1 ⊗ v2 ⊗ ...⊗ vk)
84
and ∀α < k:
σα ◦ σα+1 ◦ σα(v1 ⊗ v2 ⊗ ...⊗ vk) = (−1)|vα||vα+1|+|vα||vα+2|+|vα+1||vα+2|v1 ⊗ ...⊗ vα+2 ⊗ vα+1 ⊗ vα ⊗ ...⊗ vk
= σα+1 ◦ σα ◦ σα+1(v1 ⊗ v2 ⊗ ...⊗ vk)
Hence, the action defined above extends uniquely to an action of the symmetric group. This
action, again, commutes with the differential. We can therefore consider the subcomplex SkV ⊂T kV of invariants. The pointwise sum S(V ) :=
⊕k S
k(V ) is the free associative, supercommu-
tative algebra over V .
Proposition 23. Let V be a cochain complex of graded vector spaces. Then, there is a natural
isomorphism S(H•(V )) ∼= H•(S(V )).
Proof. By Kunneth’s theorem, we have T k(H•(V )) ∼= H•(T k(V )), for ever k ∈ N, by a natural
isomorphism
φ : [v1]⊗ [v2]⊗ ...⊗ [vk] 7→ [v1 ⊗ v2 ⊗ ...⊗ vk]
Clearly, φ commutes with the (graded) action of the symmetric group. Hence, φ descends to
a natural isomorphism Sk(H•(V )) ∼= H•(Sk(V )). Assembling these maps for every k ∈ N, we
obtain a natural isomorphism S(H•(V )) ∼= H•(S(V )), as desired.
Corollary 7. Suppose A is an associative, supercommutative algebra, B is a graded A-module,
and ∂ : A → B is a linear degree-preserving map. Suppose, in addition, that the following
conditions hold:
1. A is freely generated by a subspace V ⊂ A as an algebra over Com
2. B is freely generated over A by a subspace W ⊂ B as a module over an algebra over Com
3. ∂(V ) ⊂W
4. ∂(a1a2) = a1∂(a2) + ∂(a1)a2, ∀a1, a2 ∈ A
Then ker ∂ is the free associative, supercommutative algebra generated by ker ∂ ∩ V .
Proof. Let ∂ = ∂|V , and let C• be the following cochain complex of graded vector spaces
V∂→W → 0→ 0→ ...
85
S(C•) is the cochain complex
S(V )S(∂)−→
∞⊕n=1
(n−1⊕k=0
⊗kV ⊗W ⊗⊗n−k−1V
)Sn−→ ...
By 1, there is an algebra isomorphism p0 : S(V )∼→ A which restricts to the identity on V , and
by 2, there is a A-module isomorphism p1 :⊕∞
n=1
(⊕n−1k=0 ⊗kV ⊗W ⊗⊗n−k−1V
)Sn ∼→ B which
restricts to the identity on W .The diagram
S(V )⊕∞
n=1
(⊕n−1k=0 ⊗kV ⊗W ⊗⊗n−k−1V
)Sn
A B
S(∂)
p0 p1
∂
commutes trivially on V . Commutativity on the full symmetric algebra is then established by
induction. Indeed, if the diagram commutes for a, b ∈ S(v) then
p1(S(∂)(a� b)) = p1(S(∂)(a)� b+ a� S(∂)(b))
= p1(S(∂)(a))p0(b) + p0(a)p1(S(∂)(b))
= ∂(p0(a))p0(b) + p0(a)∂(p0(b))
= ∂(p0(a� b))
Applying Proposition (23) in homological degree 0 to the complex S(C•), we get kerS(∂) =
S(ker ∂). Therefore, ker ∂ = p0(kerS(∂)) = p0(S(ker ∂)), as desired.
We have the following obvious analogues of proposition 23 and corollary 7 in the twisted context:
Proposition 24. Let V be a cochain complex of symmetric collections. Then, there is a natural
isomorphism S(H•(V )) ∼= H•(S(V )).
Corollary 8. Suppose A is an associative, supercommutative twisted algebra, B is a twisted
A-module, and ∂ : A → B is a morphism of symmetric collections. Suppose, in addition, that
the following conditions hold:
1. A is freely generated by a subcollection V ⊂ A as a twisted algebra over Com
86
2. B is freely generated over A by a subcollection W ⊂ B as a twisted module over an algebra
over Com
3. ∂(V ) ⊂W
4. ∂(a1a2) = a1∂(a2) + ∂(a1)a2, ∀a1, a2 ∈ A
Then ker ∂ is the free twisted associative, supercommutative algebra generated by ker ∂ ∩ V .
18 Chevalley-Eilenberg Cohomology
Consider a Lie algebra h and a h-module W .
Definition 49. The Chevalley-Eilenberg cohomology H•(h,W ) of h with coefficients in W is
the cohomology of the cochain complex (C•(h,W ), δCE) where
Cn(h,W ) := Lin (∧nh,W )
and
(δf)(h1 ∧ ... ∧ hn) :=∑
1≤i≤n+1
(−1)i+1hif(h1 ∧ ... ∧ hi ∧ ... ∧ hn+1)
+∑
1≤i≤j≤n+1
(−1)i+jf([hi, hj ] ∧ h1 ∧ ... ∧ hi ∧ ... ∧ hj ∧ ... ∧ hn+1)
Suppose H o G is a Lie group and W is a H o G-module. The action of H ⊂ H o G on W
induces an infinitesimal action of the Lie algebra h on W . We can therefore consider the cochain
complex (C•(h,W ), δCE). Since G acts on H by group homomorphisms, the identity element
1 ∈ H is fixed by G. Thus, by differentiating, we obtain a linear action of G on h. On the
other hand, G acts on W via the inclusion G ⊂ H o G. Let G act on the nth component
Cn(h,W ) = Lin (∧nh,W ) by “conjugation”:
(g · f)(h1 ∧ ... ∧ hn) = gf(g−1h1 ∧ ... ∧ g−1hn)
Proposition 25. For every component Cn(h,W ), let CnG(h,W ) denote the subspace of G-
invariants. The restrictions δCE : Cn(h,W )→ Cn+1(h,W ) are well-defined for every n.
87
Proof. We want to show that the differential δCE : Cn(h,W )→ Cn+1(h,W ) commutes with the
action of G for every integer n. Let g ∈ G, f ∈ Lin (∧nh,W ) = Cn(h,W ), and h1, ..., hn+1 ∈ h.
We have
(δCE(g · f))(h1 ∧ ... ∧ hn+1)
=∑
1≤i≤n+1
(−1)i+1hi
(gf(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1)
)+
∑1≤i≤j≤n+1
(−1)i+jgf(g−1[hi, hj ] ∧ g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hj ∧ ...g−1 ∧ hn+1)
Let exp : h→ H denote the exponential map. Notice for the first summand that
hi
(gf(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1)
)=
d
dt|0 exp(thi)
(gf(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1)
)=
d
dt|0 g
((g−1exp(thi)g)f(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1)
)= g
((g−1hi)f(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1)
)For the second summand, notice that g−1[hi, hj ] = [g−1hi, g
−1hj ], since the derivative at the
identity of a Lie group homomorphism is a morphism of Lie algebras. Hence,
(δCE(g · f))(h1 ∧ ... ∧ hn+1)
=∑
1≤i≤n+1
(−1)i+1g(
(g−1hi)f(g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hn+1))
+∑
1≤i≤j≤n+1
(−1)i+jgf([g−1hi, g−1hj ] ∧ g−1h1 ∧ ... ∧ g−1hi ∧ ... ∧ g−1hj ∧ ...g−1 ∧ hn+1)
= g · (δf)(h1 ∧ ... ∧ hn+1)
Thus, δCE commutes with the action of G and, therefore, takes G-invariants to G-invariants.
It is therefore sensible to consider the subcomplex C•G(h,W ) ⊂ C•(h,W ) of G-invariants. We
denote the cohomology of this subcomplex by H•G(h,W ).
88
If W is a graded vector space and the h-action is degree-preserving, C•(h,W ) is a cochain complex
of graded vector spaces and the obvious analogue of proposition 25 holds. In our case, W is in fact
a symmetric collection {W [k]}k∈N. h acts on each W [k] by an action which commutes with the
action of Sk. The definitions of C•(h,W ) and δCE in the twisted context are virtually identical:
Cn(h,W ) is the symmetric collection defined component-wise by
Cn(h,W )[k] = Lin (∧nh,W [k])
Likewise, δnCE is a morphism of symmetric collections with kth component equal to the usual
Chevalley-Eilenberg differential Cn(h,W [k]) → Cn+1(h,W [k]). Clearly, proposition 25 holds in
the twisted case as well.
Now, consider the Lie group G∞m = NGL∞m oGL(V ) of infinity jets of origin-preserving local dif-
feomorphisms. The symmetric collection W of differential operators on alternating multi-vector
fields is a module over G∞m . In light of proposition 25, we may consider the cochain complex
C•GL(V )(ngl∞m ,W) of symmetric collections and the subcomplex C•GL(V )(ngl
∞m ,W(α1, ..., αn)) for
natural numbers α1, ..., αn.
Proposition 26. Restriction to the standard fiber induces isomorphisms
Natfieldsm (α1, ..., αn) ∼= H0
GL(V ) (ngl∞m ,W(α1, ..., αn))
for natural numbers α1, ..., αn and
Natfieldsm
∼= H0GL(V ) (ngl∞m ,W)
Proof. This is simply a matter of parsing definitions. The first two components of the cochain
complex C•GL(V ) (ngl∞m ,W) are the vector spaces
C0GL(V ) (ngl∞m ,W) = WGL(V )
C1GL(V ) (ngl∞m ,W) = LinGL(V ) (ngl∞m ,W)
Reflecting on the definition of the Chevalley-Eilenberg differential, we see that the map δCE :
WGL(V ) → LinGL(V ) (ngl∞m ,W) is given by (δCEf)(g) = gf for f ∈ WGL(V ) and g ∈ ngl∞m .
Hence,
89
H0GL(V ) (ngl∞m ,W) = ker δ : WGL(V ) → LinGL(V ) (ngl∞m ,W)
= {f ∈WGL(V ) | gf = 0 ∀g ∈ ngl∞m}
Since the Lie group NGL∞m is both connected and simply connected,
{f ∈WGL(V ) | gf = 0 ∀g ∈ ngl∞m} =(WGL(V )
)NGL∞mBy proposition 16, restriction to the standard fiber defines a linear isomorphism Natfields
m∼=(
WGL(V ))NGL∞m . Thus, Natfields
m∼= H0
GL(V ) (ngl∞m ,W), as desired. The proof of the first iso-
morphism is absolutely analogous, replacing W above with W(α1, ..., αn) and Natfieldsm with
Natfieldsm (α1, ..., αn).
The map f defined in section 11 gives a correspondence between a particular space of rooted
multigraphs and C0GL(V ) (ngl∞m ,W) = WGL(V ). We wish to define maps analogous to f for other
components of the cochain complex C•GL(V ) (ngl∞m ,W). As a GL(V )-module
ngl∞m∼=∞⊕i=1
SiV ∗ ⊗ V
Hence,
CnGL(V ) (ngl∞m ,W) = LinGL(V ) (∧nngl∞m ,W)
∼=⊕
i1,...,in∈NLinGL(V )
n∧j=1
SijV ∗ ⊗ V
,W
∼=
⊕i1,...,in∈N
n∧j=1
SijV ⊗ V ∗⊗W
GL(V )
We define a symmetric collection of rooted multigraphs which, for stable values of m, corre-
sponds isomorphically to CnGL(V ) (ngl∞m ,W). Note: a bit of extra care is required for n > 1 to
handle the wedge product in the vector space above. This is managed by making suitable identifi-
cations in the corresponding space of rooted multigraphs, as described in the following definition.
90
Definition 50. For each n ≥ 0 and k ≥ 1, Let Gran[k] denote the vector space spanned by
finite ordered multigraphs containing
1. k ordinary vertices • labeled 1, ..., k with symmetric inputs and alternating outputs
2. n white vertices ◦ labeled 1, ..., n, each with exactly 1 output and at least 2 symmetric
inputs
3. 1 root vertex � with symmetric inputs and no outputs
Each Gran[k] is graded by the valence of the root and Sk acts by relabeling ordinary vertices.
Further, we make the following identifications
1. For multigraphs G and G′ in Gran[k] that differ only by an even permutation of the white
vertex labelings, we put G = G′
2. For multigraphs G and G′ in Gran[k] that differ only by an odd permutation of the white
vertex labelings, we put G = −G′
Gran
is the symmetric collection {Gran[k]}k∈N. The subcollections Gran,irred
, Gran
acyc and
Gran,irred
acyc are defined analogously to their degree-0 counterparts.
Through a procedure absolutely analogous to the construction of f in section 11, we define an
epimorphism fn
: Gran → CnGL(V ) (ngl∞m ,W) for each natural number n. In lieu of explicit defi-
nitions, we present several illustrative examples. For explicit definitions (in the slightly simpler,
but entirely analogous vector field case), see [16].
Example 34. Gra1(1, 1) is a four-dimensional vector space spanned by the rooted multigraphs
•1
•2
◦ 1
�
,•1
•2
◦ 1�
,•2
•1
◦ 1�
,
•1
•2
◦ 1
�
The first three basis elements map under f1
to elementary invariant tensors in the vector space
LinGL(V )
(S2V ⊗ V ∗, V ∗ ⊗ V ∗ ⊗ V
)and the fourth basis element maps to an elementary in-
variant tensor in LinGL(V )
(S3V ⊗ V ∗, V ∗ ⊗ V ∗
). Both of these vector spaces are canonically
regarded as subspaces of C1GL(V ) (ngl∞m ,W).
Example 35. Gra1(2) is a one-dimensional vector space spanned by the multigraph
91
•1
◦ 1 �
This multigraph is mapped by f1
to an elementary invariant tensor in LinGL(V )
(S2V ⊗ V ∗,∧2V ∗ ⊗ V
),
again, regarded as a subspace of C1GL(V ) (ngl∞m ,W).
Example 36. Gra2(1, 1) is a three-dimensional vector space spanned by the multigraphs
•1
•2
◦1
◦2
�
,
•1
•2
◦1
◦2
�
,
•1
•2
◦1
◦2
�
Note that swapping the two white vertices in any of these three basis elements produces a minus
sign. As does swapping the two ordinary vertices in the first two multigraphs. Transposing the
two ordinary vertices in the third multigraph has no effect. All three multigraphs are mapped
by f2
to elementary invariant tensors in LinGL(V )
(∧2(S2V ⊗ V ∗), V ∗ ⊗ V ∗ ⊗ V
), regarded as a
subspace of C2GL(V ) (ngl∞m ,W).
Remark 5. A key fact about fn
is that it restricts to an isomorphism fn
: Gran(α1, ..., αk)
∼→CnGL(V ) (ngl∞m ,W(α1, ..., αk)) for any collection of α1, ..., αk with the property that n+
∑ki=1 αi <
m (note: this is analogous to part 3 of proposition 17, with the additional n on the left coming
from the n outgoing edges from the n white vertices). The proof of this fact is a straightforward
application of the Invariant Tensor Theorem.
Next, we define a differential δ on the graph complex Gra•
corresponding to the Chevalley-
Eilenberg differential on the cochain complex C•GL(V ) (ngl∞m ,W). We will define δ as a sum
δ1 + δ2 and define each piece separately.
Definition 51. Let G be a rooted multigraph in Gran[k]. Given any i, 1 ≤ i ≤ k, construct the
multigraph δ1,i(G) via the local replacement rule
92
δ1,i : •i
· · ·
· · ·
v
w
7−→∑
s+u=v+1s≥2
•i
◦
· · ·
· · ·· · ·
( )( ) alt
sym
u
s
w
−◦
• i
· · ·
· · ·
· · ·
( )sym
s
u
w
where “sym” denotes the symmetrization, obtained by summation over all (u, s)-unshuffles, and
“alt” denote the alternation, obtained by summation over all (w, 1) unshuffles, with signs deter-
mined by the signs of the permutations. The new white vertex introduced in the replacement
rule above is labeled 1 in δ1,i(G) and all other white vertices are labeled so as to retain the
original ordering on white vertices in G (i.e. white vertex j is relabeled j+1 for all j, 1 ≤ j ≤ n).
We then define δ1 as a sum over all ordinary vertices:
δ1(G) :=
k∑i=1
δ1,i(G)
It is clear that δ1 defines a morphism from Gran
to Gran+1
for every n.
Definition 52. Let G be a rooted multigraph in the symmetric collection Gran
and 1 ≤ i ≤ n.
Construct the multigraph δ2,i(G) via the local replacement rule
δ2,i :◦i
· · ·
v
7−→∑
u+s=v+1s,u≥2
◦i
◦ i+ 1
· · ·
· · ·( )sym
s
u
In the multigraph δ2,i(G), relabel each white vertex j > i as j+ 1 and retain the labelings on all
white vertices j < i. We then define δ2 as a sum over all white vertices:
93
δ2(G) :=
n∑i=1
(−1)i+1δ2,i(G)
It is similarly clear that δ2 defines a morphism from Gran
to Gran+1
.
Definition 53. δ := δ1 + δ2
We give some simple examples in low cohomological degree.
Example 37.
δ
•1
�
= 0
Example 38.
δ
•1
•2
� =
•1
◦1
• 2
�
Example 39.
δ
•2
◦1 • 4
•1
•3
�
=
•2
◦2 ◦1
•1
•3
•4
�
−
•2
◦2 ◦1
•1
•3
•4
�
−
•2
◦2 ◦ 1
•1
•3
�• 4
δ1
+
•2
◦1 • 4
•1
•3
◦2
�
+
•2
◦1 • 4
•1
•3
◦ 2
�
+
•2
◦1
• 4
•1
•3
◦2
�
δ2
It is clear from the definition of δ that the following diagram commutes:
94
Gra Gra1
Gra2 · · ·
C0GL(V ) (ngl∞m ,W) C1
GL(V ) (ngl∞m ,W) C2GL(V ) (ngl∞m ,W) · · ·
δ
f
δ
f1
δ
f2
δCE δCE δCE
Indeed, δ1 is precisely the graphical representation of the first part of the Chevalley-Eilenberg
differential and δ2 is precisely the graphical representation of the second. In particular, for natural
numbers α1, ..., αn satisfying 1 +∑ni=1 αi < m, f restricts to an isomorphism Gra(α1, ..., αn)
∼→C0GL(V ) (ngl∞m ,W(α1, ..., αn)) (see part 3 of proposition 17) and f
1restricts to an isomorphism
Gra1(α1, ..., αn)
∼→ C1GL(V ) (ngl∞m ,W(α1, ..., αn)) (see remark 5). In this case, the first square in
the diagram above restricts to:
Gra(α1, ..., αn) Gra1(α1, ..., αn)
C0GL(V ) (ngl∞m ,W(α1, ..., αn)) C1
GL(V ) (ngl∞m ,W(α1, ..., αn))
∼ ∼
δ
δCE
f f1
Hence, H0GL(V ) (ngl∞m ,W(α1, ..., αn)) = f
(ker δ : Gra(α1, ..., αn)→ Gra
1(α1, ..., αn)
). But by
proposition 26, H0GL(V ) (ngl∞m ,W(α1, ..., αn)) ∼= Natfields
m (α1, ..., αn). Thus, we have shown:
Theorem 10. For natural numbers α1, ..., αn satisfying 1+∑ni=1 αi < m, invariant-theoretic re-
duction and the correspondence outlined in section 11 between invariant tensors and multigraphs
induce a linear isomorphism
Natfieldsm (α1, ..., αn) ∼= ker δ : Gra(α1, ..., αn)→ Gra
1(α1, ..., αn)
Proposition 27. The morphism δ : Gra→ Gra1
satisfies the hypotheses of corollary 8.
Proof. From proposition 15, we have
Gra ∼= S(Gra
irred)
It is equally clear that the wedge product turns Gra1
into a twisted Gra-module which is freely
generated (as a twisted module over an algebra over Com) by the subcollection Gra1,irred
. In-
deed, every multigraph G ∈ Gra1decomposes as a wedge product of irreducible pieces, precisely
one of which is contained in Gra1,irred
, and this decomposition is unique up to re-ordering
95
of pieces. Thus, hypotheses 1 and 2 are satisfied. For 3, consider an irreducible multigraph
G ∈ Gra1,irred. Hence, G−� is connected. Since the replacement rules defining δ are connected,
δ(G) − � = δ (G−�) is connected, and therefore, δ(G) is irreducible, which establishes 3. For
4, notice that δ2 is 0 on Gra. Hence,
δ(G1 ∧G2) = δ1(G1 ∧G2) + δ2(G1 ∧G2)
=∑
v∈G1tG2
δ1,v(G1 ∧G2)
=∑v∈G1
δ1,v(G1 ∧G2) +∑v∈G2
δ1,v(G1 ∧G2)
=∑v∈G1
δ1,v(G1) ∧G2 +∑v∈G2
G1 ∧ δ1,v(G2)
=
(∑v∈G1
δ1,v(G1)
)∧G2 +G1 ∧
(∑v∈G2
δ1,v(G2)
)= δ(G1) ∧G2 +G1 ∧ δ(G2)
which establishes 4.
It follows from corollary 8 that every element of ker δ : Gra→ Gra1
is a linear combination of
wedge products of irreducible kernel elements. Symbolically,
ker δ : Gra→ Gra1 ∼= S
(ker δ : Gra
irred → Gra1)
Furthermore, by a simple graph-theoretic argument, we see that all kernel elements are internally
acyclic:
Theorem 11.(
ker δ : Gra→ Gra1)⊂ Graacyc
We begin by establishing a bit of extra notation. Let Gracyc ⊂ Gra denote the subcollec-
tion spanned by internally cyclic multigraphs (i.e. multigraphs G with G − � cyclic) so that
Gra ∼= Graacyc ⊕ Gracyc. For each n, k ∈ N, let Gracyc[n, k] ⊂ Gracyc[n] denote the subspace
spanned by multigraphs with n ordinary vertices and k ordinary vertices contained in an internal
cycle. Similarly, we can define Gra1
cyc and Gra1
cyc[n, k], with Gra1
cyc[n, k] ⊂ Gra1
cyc[n] being the
subspace spanned by multigraphs with n ordinary vertices, 1 white vertex, and k vertices (of
either type) contained in an internal cycle.
We want to show that the restriction
96
δ : Gracyc → Gra1
cyc
is injective, i.e that
δ[n] : Gracyc[n]→ Gra1
cyc[n]
is injective for every n ∈ N. Fix a particular n ∈ N. It is clear from the definition of δ that
δ[n] = δ+0[n] + δ+1[n]
where δ+0[n] is the component of δ[n] : Gracyc[n]→ Gra1
cyc[n] which fixes the number of vertices
contained in an internal cycle, and δ+1[n] is the component which increases this number by one.
In this notation, we have the following simple criterion.
Lemma 3. δ[n] : Gracyc[n] → Gra1
cyc[n] is injective if δ+0[n] : Gracyc[n, k] → Gra1
cyc[n, k] is
injective for every k ∈ N.
Proof. Suppose δ[n] : Gracyc[n]→ Gra1
cyc[n] has nontrivial kernel. Hence, there is a multigraph
x ∈ Gracyc[n] not equal to 0 such that δ[n](x) = 0. x can be decomposed as a finite sum
x = xk + xk+1 + ...+ xk+a
where xi ∈ Gracyc[n, i] for i = k, k+ 1, ..., k+a and xk 6= 0. Fix a basis of multigraphs for Gra1.
Our particular choice of basis induces a projection
π : Gra1
cyc[n]� Gra1
cyc[n, k]
Since δ[n] cannot reduce the number of cycle vertices in a multigraph, we have
0 = π ◦ δ[n](x) = δ+0[n](xk)
Yet, xk 6= 0 be assumption, so δ+0[n] : Gracyc[n, k]→ Gra1
cyc[n, k] cannot be injective.
Proof of theorem 11. Consider the subspace Gra1
cyc[n, k]′ ⊂ Gra1
cyc[n, k] spanned by multigraphs
with one white vertex contained in an internal cycle such that precisely one input to this vertex
does not lie on an internal cycle. Fixing a basis of multigraphs for Gra1, we obtain a projection
97
π : Gra1
cyc[n, k]� Gra1
cyc[n, k]′
and contracting along the unique input not contained in a cycle, we obtain a linear map
r : Gra1
cyc[n, k]′ → Gracyc[n, k]
We claim r ◦ π ◦ δ+0[n] = w id where w is some nonzero integer. From this, the injectivity of
δ+0[n] follows trivially, and by lemma 3, we are done. Select a multigraph G ∈ Gracyc[n, k] and
consider a particular (internal) cycle C ⊂ G. Vertices in C are of three possible types:
1. v has two inputs contained in C
2. v has both an input and an output contained C
3. v has two outputs contained in C
Clearly, C contains at least one vertex of type 1 or 2. Denote the number of such vertices by wC
and let w be the sum∑wC over all internal cycles in G. Notice:
98
δ
•
= ◦•
+ ...
δ
•
= ◦•
+ ◦
•
+ ◦•
− •
◦
+ ...
δ
•
= ◦•
+ ◦•
+ ...
Figure 1: Replacement Rules for Type 1 Vertices
δ
•
= ◦• + ◦
•
+ ◦•
− ◦• + ...
Figure 2: Replacement Rules for Type 2 Vertices
99
δ
•
= 0 + ...
δ
•
= ◦
•
+ ◦•
+ ...
Figure 3: Replacement Rules for Type 3 Vertices
From these illustrations it is clear that r ◦ π ◦ δ+0[n] = w id, as desired.
References
[1] R. S. Palais and C.-L. Terng, “Natural Bundles Have Finite Order,” Topology, 1976.
[2] C.-L. Terng, “Natural Vector Bundles and Natural Differential Operators,” American Jour-
nal of Mathematics, 1978.
[3] D. B. A. Epstein and W. P. Thurston, “Transformation Groups and Natural Bundles,” Proc.
London Math. Soc., vol. 3, no. 38, 1979.
[4] R. S. Palais, “Natural Operations on Differential Forms,” Transactions of the American
Mathematical Society, 1959.
[5] P. W. M. Ivan Kolar and J. Slovak, Natural Operations in Differential Geometry. Springer-
Verlag, 1993.
[6] F. Chapoton and M. Livernet, “Pre-Lie Algebras and the Rooted Trees Operad,” Interna-
tional Mathematics Research Notices, 2001.
[7] J.-L. Loday, Generalized Bialgebras and Triples of Operads. Astrisque, 2008.
[8] S. Merkulov, Frobenius Manifolds: Quantum Cohomology and Singularities, vol. E36 of
Aspects Math. Vieweg, 2004.
100
[9] W. M. Mikulski, “Locally Determined Associated Spaces,” J. London Math. Soc., vol. 32,
1985.
[10] M. Markl, “Lie Elements in Pre-Lie Algebras, Trees, and Cohomology Operations,” Journal
of Lie Theory, 2006.
[11] M. Markl, “Natural Differential Operators and Graph Complexes,” Differential Geometry
and its Applications, 2009.
[12] A. Joyal and R. Street, “Braided tensor categories,” Advances in Mathematics, 1993.
[13] J.-L. Loday and B. Vallette, Algebraic Operads. Springer-Verlag, 2012.
[14] M. Gerstenhaber, “The Cohomology Structure of an Associative Ring,” Annals of Mathe-
matics, vol. 78, no. 2, 1963.
[15] T. J. L. Frederick R. Cohen and J. P. May, The Homology of Iterated Loop Spaces. Springer-
Verlag, 1976.
[16] M. Markl, “Invariant Tensors and Graphs.” Preprint arXiv:0801.0418 [math.CA].
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