marina avitabile
TRANSCRIPT
Laguerre Polynomials of Derivations
Marina Avitabile
Universita di Milano - Bicocca
Trento, June 2013
Joint work with Sandro Mattarei
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 1 / 25
1 Gradings of non-associative algebras
2 The exponential of a derivationThe Artin-Hasse exponential of a derivation
3 Laguerre polynomialsLaguerre polynomials modulo p
4 A model special case
5 General Case
6 Toral switching
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 2 / 25
Gradings
Definition
Let A be a (finite-dimensional) non-associative algebra over a field F. Agrading of A over an abelian group G is a direct sum decomposition:
A =⊕g∈G
Ag
such that AgAh ⊆ Ag+h.
A derivation D of A is a linear map D : A→ A such that
D(a · b) = D(a) · b + a · D(b)
for all a, b ∈ A.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 3 / 25
Gradings
Let D be a derivation of A, with all its characteristic roots in F. Thedirect sum decomposition of A into (generalized) eigenspaces for D
A =⊕α∈F
Aα
(where Aα = {x ∈ A : (D − α id)i (x) = 0, for some i > 0}) is a grading ofA over (F,+), i.e. AαAβ ⊆ Aα+β.Let σ be an automorphism of A, with all its characteristic roots in F. Thedirect sum decomposition of A into generalized eigenspaces for σ
A =⊕α∈F
Aα
is a grading of A over (F∗, ·), i.e. AαAβ ⊆ Aαβ.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 4 / 25
The exponential of a derivation
Let char(F) = 0 and D be a nilpotent derivation of A, with Dn = 0.The exponential map
exp(D) =n−1∑i=0
D i
i !
defines an automorphism of A.
D ◦m = m ◦ (D ⊗ id + id⊗D), where m : A⊗ A→ A is themultiplication map.
Set X = D ⊗ id and Y = id⊗D, then exp(X + Y ) = exp(X ) · exp(Y )(if X and Y commute).
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 5 / 25
The exponential of a derivation
From now on assume char(F) = p > 0
S. MattareiArtin-Hasse exponentials of derivationsJ. Algebra 294 (2005), 1–18
Let D be a nilpotent derivation of A with Dp = 0, then
exp(D) =
p−1∑i=0
D i
i !
and it defines a bijective linear map on A.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 6 / 25
The exponential of a derivation
Define the truncated exponential of D as
E (D) =
p−1∑i=0
D i
i !.
Direct computation shows that
E (D)x · E (D)y − E (D)(xy) =
2p−2∑t=p
p−1∑i=t+1−p
(D ix)(Dt−iy)
i !(t − i)!.
If p is odd and Dp+1
2 = 0 then each term in the sum vanishes and exp(D)is an automorphism of A, but in general E (D) it is not an automorphismof A even if Dp = 0.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 7 / 25
The exponential of a derivation
However, under certain hypotheses, the (truncated) exponential of aderivation has the property of sending a grading of A into another gradingof A.
Lemma (S. Mattarei)
Let A be a non-associative algebra over a field of positive characteristic p,with derivation D such that Dp = 0. Then
exp(D)x · exp(D)y − exp(D)(xy) = exp(D)
(p−1∑i=1
(−1)i
iD ix · Dp−iy
)
for all x , y ∈ A.
Let A = ⊕Ai be a grading of A over the integers modulo m. A derivationD of A is graded of degree d if D(Ai ) ⊆ Ai+d for every i .
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 8 / 25
The exponential of a derivation
Theorem (S. Mattarei)
Let A = ⊕Ai be a non-associative algebra over a field of positivecharacteristic p, graded over the integers modulo m. Suppose that A has agraded derivation D of degree d, with m | pd, such that Dp = 0. Thenthe direct sum decomposition A = ⊕ exp(D)Ai is a grading of A over theintegers modulo m.
Proof.
Verify that exp(D)As · exp(D)At ⊆ exp(D)As+t .Let x ∈ As and y ∈ At , then D i (x) ∈ As+di and Dp−i (y) ∈ At+d(p−i) and
D i (x) · Dp−i (y) ∈ As+t+pd = As+t . The previous lemma yields
exp(D)x · exp(D)y = exp(D)(xy)︸ ︷︷ ︸exp(D)(As+t)
+ exp(D)
(p−1∑i=0
(−1)i
iD ix · Dp−iy
)︸ ︷︷ ︸
exp(D)(As+t)
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 9 / 25
The Artin-Hasse exponential of a derivation
The assumption that Dp = 0 can be relaxed by considering theArtin-Hasse exponential series which is defined as
Ep(X ) = exp
( ∞∑i=0
X pi
pi
)=∞∏i=0
exp
(X pi
pi
)∈ Zp[[X ]].
Theorem (S. Mattarei)
Let A = ⊕Ai be a non-associative algebra over a field of positivecharacteristic p, graded over the integers modulo m. Suppose that A has anilpotent graded derivation D of degree d, with m | pd. Then the directsum decomposition A = ⊕Ep(D)Ai is a grading of A over the integersmodulo m.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 10 / 25
The Artin-Hasse exponential of a derivation
Ideas from the proof.
Let D be a nilpotent derivation of A. Then there exist integers ai ,j , withai ,j = 0 when p 6 | i + j such that
Ep(D)x · Ep(D)y − Ep(D)(xy) = Ep(D)
∞∑i ,j=1
ai ,jDix · D jy
for all x , y ∈ A.Set X = D ⊗ id and Y = id⊗D, then the identity above is equivalent to
(Ep(X + Y ))−1Ep(X ) · Ep(Y ) = 1 +∞∑
i ,j=1
ai ,jXiY j
in Q[[X ,Y ]], where ai ,j = 0 when where p 6 | i + j .
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 11 / 25
Laguerre polynomials
The classical (generalized) Laguerre polynomial of degree n ≥ 0 is definedas
L(α)n (X ) =
n∑k=0
(α + n
n − k
)(−X )k
k!
where α is a parameter, classically in the complex numbers. We may also
view L(α)n (x) as a polynomial in two indeterminates X and α.
Recurrence relations
1 L(γ)n (X ) = L
(γ+1)n (X )− L
(γ+1)n−1 (X )
2 nL(γ+1)n (X ) = (n − X )L
(γ+1)n−1 (X ) + (n + γ)L
(γ)n−1(X )
3ddX L
(γ)n (X ) = −L
(γ+1)n−1 (X ) = L
(γ)n (X )− L
(γ+1)n (X )
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 12 / 25
Laguerre polynomials modulo p
Let p be a fixed prime, we consider Laguerre polynomials of degree p − 1
L(α)p−1(X ) =
p−1∑k=0
(α + p − 1
p − 1− k
)(−X )k
k!
≡(α + p − 1
p − 1
) p−1∑k=0
X k
(α + k)(α + k − 1) · · · (α + 1)mod p
because(p−1
k
)≡ (−1)k mod p.
Special case α = 0,
L(0)p−1(X ) ≡ E (X ) =
p−1∑k=0
X k
k!mod p.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 13 / 25
Laguerre polynomials modulo p
We have
pL(γ)p (X ) = p
p∑k=0
(γ + 1− p
p − k
)(−X )k
k!
≡ X p − (γp − γ) mod p
Crucial congruence
Xd
dXL
(γ)p−1(X ) ≡ (X − γ)L
(γ)p−1(X ) + X p − (γp − γ) mod p
Special case γ = 0,
XE ′(X ) ≡ XE (X ) + X p mod p
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 14 / 25
Laguerre polynomials modulo p
Lemma
We have L(Zp)p−1 (Zp − Z ) =
p−1∏i=1
(1 + Z/i)i in Fp[Z ].
Sketch of the proof.
Laguerre polynomials satisfy
(Zp − Z )L(Zp+1)p−1 (Zp − Z ) = ZpL
(Zp)p−1 (Zp − Z ) (1)
Since L(0)(0) = 1 we can write L(Zp)p−1 (Zp − Z ) =
s∏i=1
(1− Z/αi ) in Fp[Z ].
Thenp−1∏j=1
(Z − j)s∏
i=1
(Z − (αi − 1)) = Zp−1s∏
i=1
(Z − αi )
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 15 / 25
Sketch of the proof.
Let α ∈ Fp be a root of L(Zp)p−1 (Zp − Z ) with multiplicity m, then
1 if α = 0 then α + 1 is a root with multiplicity m + p − 1
2 if α ∈ F∗p then α + 1 is a root with multiplicity m − 1
3 if α 6∈ Fp then α + 1 is a root with multiplicity m.
Since 0 is not a root, it follows that the elements of F∗p are roots of
L(Zp)p−1 (Zp − Z ) with the claimed multiplicities.
There are no further roots.
1 ∂L(Zp)p−1 (Zp − Z ) ≤ p(p − 1) and ∂
∏p−1i=1 (1 + Z/i)i = p(p−1)
2
2 ZpL(Zp)p−1 (Zp − Z )L
(−Zp)p−1 (−Zp + Z ) is invariant under the substitution
Z → Z + 1
3 ZpL(Zp)p−1 (Zp − Z )L
(−Zp)p−1 (−Zp + Z ) has zero derivative
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 16 / 25
Sketch of the proof.
Therefore ZpL(Zp)p−1 (Zp − Z )L
(−Zp)p−1 (−Zp + Z ) is a polynomial in Zp2 − Zp.
Since its degree cannot exceed 2p2 − p then it cannot exceed p2. This
proves that ∂L(Zp)p−1 (Zp − Z ) ≤ p(p−1)
2 and completes the proof.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 17 / 25
An exponential-like property of Laguerre polynomials
We now turn the modular differential equation into an analougue of thefunctional equation exp(X ) exp(Y ) = exp(X + Y ) for the classicalexponential
Proposition
Consider the subring R = Fp[α, β, ((α + β)p−1 − 1)−1] of the ringFp(α, β) of rational expressions in the indeterminates α and β, and let Xand Y be further indeterminates. Then there exists rational expressionsci (α, β) ∈ R such that
L(α)p−1(X )L
(β)p−1(Y ) ≡ L
(α+β)p−1 (X + Y )
(c0(α, β) +
p−1∑i=1
ci (α, β)X iY p−i
)
in R[X ,Y ], modulo the ideal generated by X p − (αp − α) andY p − (βp − β).
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 18 / 25
An exponential-like property of Laguerre polynomials
Special case: α = 0 = β, L(0)p−1(X ) = E (X )
E (X ) · E (Y ) ≡ E (X + Y )
(1 +
p−1∑i=1
(−1)iX iY p−i/i
)
in Fp[X ,Y ] modulo the ideal generated by X p and Y p.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 19 / 25
A model special case
Theorem
Let A =⊕
k Ak be a Z/mZ-grading of A;
let D ∈ Der(A), graded of degree d, with m | pd, such thatDp2
= Dp;
let A =⊕
a∈FpA(a) be the decomposition of A into generalized
eigenspaces for D;
assuming Fpp ⊆ F, fix γ ∈ F with γp − γ = 1;
let LD : A→ A be the linear map on A whose restriction to A(a)
coincides with L(aγ)p−1(D).
Then A =⊕
k LD(Ak) is a Z/mZ-grading of A.
Sketch of the proof.
The linear map LD is bijective: (LD)p acts on A(a) as multiplication bythe nontrivial scalar
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 20 / 25
A model special case
Sketch of the proof.
L((aγ)p)p−1 (Dp) = L
((aγ)p)p−1 (a) = L
((aγ)p)p−1 ((aγ)p − (aγ)).
The direct sum decomposition A =⊕
k LD(Ak) is a grading of A over theintegers modulo m.
Dp(Ak) ⊆ Ak , Ak = ⊕a∈FpAk ∩ A(a);
let x ∈ Ak ∩ A(a) and y ∈ Al ∩ A(b);
for any θ ∈ F, L(θ)p−1(D) ◦m = m ◦ L
(θ)p−1(D ⊗ id + id⊗D).
The proposition yields
LDx · LDy = LD
(c0(aγ, bγ)xy +
p−1∑i=1
ci (aγ, bγ)D ix · Dp−iy
)
thus LDAk · LDAl ⊆ LDAk+l .
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 21 / 25
General Case
Theorem
Let A =⊕
k Ak be a Z/mZ-grading of A;
let D ∈ Der(A), graded of degree d, with m | pd, such that Dpr isdiagonalizable over F ;
let A =⊕
ρ∈F A(ρ) be the decomposition of A into generalizedeigenspaces for D;
assuming F large enough, there is a p-polynomial g(T ) ∈ F [T ], such
that g(D)p − g(D) = Dpr , set h(T ) =∑r−1
i=1 T pi ;
let LD : A→ A be the linear map on A whose restriction to A(ρ)
coincides with L((g(ρ)−h(D))p−1 (D).
Then A =⊕
k LD(Ak) is a Z/mZ-grading of A.
On the subalgebra ker(Dpr ) the map LD coincides with the Artin-Hasseexponential series.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 22 / 25
General Case
Set
α = α− h(X ) = α− (X p + X p2+ · · ·X pr−1
)
β = β − h(Y ) = β − (Y p + Y p2+ · · ·Y pr−1
)
thus α + β = α + β − h(X + Y ).We have
L(α)p−1(X )L
(β)p−1(Y ) ≡ L
(α+β)p−1 (X + Y ) ·
(c0(α, β) +
p−1∑i=1
ci (α, β)X iY p−i
)
modulo the ideal (X p − (αp − α),Y p − (βp − β)), that is the ideal
generated by (X pr − (αp − α),Y pr − (βp − β)).
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 23 / 25
Toral switching
Replace a torus T of a restricted Lie algebra L with another torus Tx , byapplying to T a sort of exponential in the inner derivation ad x , for acertain root vector x ∈ L. This techinique goes back to Winter and it hasbeen generalized by Block, Wilson and Premet. A crucial step in the toralswitching process is the keep control on the root space decomposition withrespect to the new torus, by constructing a linear map E (x , λ) mappingthe root spaces with respect to T bijectively onto the root spaces withrespect to Tx .The linear map E (x , λ) coincides with our map LD and the toral switchingprocess, a part for the strictly Lie-theoretic aspects, can be viewed as aspecial instance of our theorem in the general case.
Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 24 / 25
Grading switching
Grading switching
applies to nonassociative algebras;
is not restricted to gradings over groups of exponent p.
M. Avitabile and S. MattareiLaguerre polynomials of derivationssubmitted (arXiv:1211.4432)
It finds one application (to thin Lie algebras) in
M. Avitabile and S. MattareiNottingham Lie algebras with diamonds of finite and infinite typesubmitted (arXiv:1211.4436)
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