pseudo h-type algebras, integer structure constants and...
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Pseudo H-type algebras, integer structureconstants and isomorphisms.
Irina Markina
University of Bergen, Norway
joint work with A. Korolko, M. Godoy,
K. Furutani, A. Vasiliev, C. Autenried
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 1/27
Heisenberg algebra
H1 = span{X,Y } ⊕ span{Z} = V ⊕Z, [X,Y ] = Z
[X,Y ] = Z is unique non vanishing commutator
Let (· , ·) be an inner product such that X,Y, Z areorthonormal.
Define J : Z × V → V an operator
(J(z, v), u)V := (z, [v, u])Z = (z, adv u)Z .
for any z ∈ Z and u, v ∈ V .
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 2/27
Properties of J
(J(z, v), u)V := (z, [v, u])Z = (z, adv u)Z .
• J : Z × V → V is a bilinear map
• J is skew symmetric with respect to (· , ·)V :
(J(z, v), u)V = −(v, J(z, u))V .
• J(·, v) = ad∗v(·),
• J(·, v) = ad−1v (·), as
adv :(
ker(adv)⊥, (· , ·)V
)
↔ (Z, (· , ·)Z)
J(·, v) : (Z, (· , ·)Z) ↔(
ker(adv)⊥, (· , ·)ker(adv)⊥
)
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 3/27
Properties of J
The operators
adv :(
ker(adv)⊥, (· , ·)ker(adv)⊥
)
↔ (Z, (· , ·)Z)
J(·, v) : (Z, (· , ·)Z) ↔(
ker(adv)⊥, (· , ·)ker(adv)⊥
)
is an isometry for (v, v)V = 1
(J(z, v), J(z, v))V = (z, z)Z(v, v)V
or
(J(z,v
‖v‖), J(z,
v
‖v‖))V = (z, z)Z
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 4/27
Heisenberg type algebra
Theorem, A. Kaplan, 1981
A two step nilpotent Lie algebra
n =(
Z ⊕⊥ V, [· , ·], (· , ·) = (· , ·)Z + (· , ·)V
)
is an H-type Lie algebra if the operator
(J(z, v), v′)V =: (z, [v, v′])Z = (z, adv v′)Z
is an isometry for any unit length vector v ∈ V .
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 5/27
Heisenberg type algebra
Theorem, A. Kaplan, 1981
A two step nilpotent Lie algebra
n =(
Z ⊕⊥ V, [· , ·], (· , ·) = (· , ·)Z + (· , ·)V
)
is an H-type Lie algebra if the operator
(J(z, v), v′)V =: (z, [v, v′])Z = (z, adv v′)Z
is an isometry for any unit length vector v ∈ V .
An H-type algebra n exists iff J2(z, ·) = −(z, z)Z IdV .
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 5/27
Relation to Clifford algebras
〈J(z, v), J(z, v′)〉V = 〈z, z〉Z〈v, v′〉V . (1)
〈J(z, v), v′〉V = −〈v, J(z, v′)〉V (2)
〈J2(z, v), v′〉V = −〈J(z, v), J(z, v′)〉V = −〈z, z〉Z〈v, v′〉V
〈(−〈z, z〉Z)v, v′〉V =⇒
J2(z, v) = −〈z, z〉Zv or J2(z, ·) = −〈z, z〉Z IdV (3)
(1) + (2) =⇒ (3)
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 6/27
Relation to Clifford algebras
(1) 〈J(z, v), J(z, v′)〉V = 〈z, z〉Z〈v, v′〉V .
(2) 〈J(z, v), v′〉V = −〈v, J(z, v′)〉V
(3) J2(z, ·) = −〈z, z〉Z IdV
The first property is the composition of quadratic formsand
The last property is defining property for Cliffordalgebra.
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 7/27
Clifford algebra
Let (W, 〈· , ·〉W ) be a scalar product space.
The Clifford algebra Cl((W, 〈· , ·〉W )) is an associativealgebra with unit I, product ⊗, factorized by the relation
w ⊗ w = −〈w,w〉W I or(
w ⊗ u+ u⊗ w = −2〈w, u〉W I
)
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 8/27
Clifford algebra
Let (W, 〈· , ·〉W ) be a scalar product space.
The Clifford algebra Cl((W, 〈· , ·〉W )) is an associativealgebra with unit I, product ⊗, factorized by the relation
w ⊗ w = −〈w,w〉W I or(
w ⊗ u+ u⊗ w = −2〈w, u〉W I
)
If (w1, . . . , wn) is an orthonormal basis of W
wk ⊗ wk = −〈wk, wk〉W I, wk ⊗ wl = −wl ⊗ wk, k 6= l
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 8/27
Clifford module
The algebra homomorphism J :
J : Cl(W, 〈· , ·〉W ) → End(V )
is called representation and (V, J) is
Clifford module for Cl(W, 〈· , ·〉W )
w 7→ J(w, ·) : V → Vw ⊗ w 7→ J ◦ J = J2(w, ·) : V → V
−〈w,w〉W I 7→ −〈w,w〉W IdV
=⇒ J2(w, ·) = −〈w,w〉W IdV
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 9/27
List of Clifford algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 10/27
Relation to Clifford module
(1) 〈J(z, v), J(z, v′)〉V = 〈z, z〉Z〈v, v′〉V .
(2) 〈J(z, v), v′〉V = −〈v, J(z, v′)〉V
(3) J2(z, ·) = −〈z, z〉Z IdV
Question: given (3) can we construct a general H-typealgebra?
(N = V ⊕⊥ Z, [· , ·], 〈· , ·〉)
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 11/27
Relation to Clifford module
(1) 〈J(z, v), J(z, v′)〉V = 〈z, z〉Z〈v, v′〉V .
(2) 〈J(z, v), v′〉V = −〈v, J(z, v′)〉V
(3) J2(z, ·) = −〈z, z〉Z IdV
Question: given (3) can we construct a general H-typealgebra?
(N = V ⊕⊥ Z, [· , ·], 〈· , ·〉)
Answer: yes if we add (1) or (2)
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 11/27
Pseudo H-type algebras
Let (N = V ⊕⊥ Z, [· , ·], 〈· , ·〉) be a two step nilpotent Liealgebra such that
(Z, 〈· , ·〉Z ), (V, 〈· , ·〉V ) are non degenerate
The Lie algebra N is a called pseudo H-type Liealgebra if the operator
〈J(z, v), v′〉V =: 〈z, [v, v′]〉Z = 〈z, adv v′〉Z
satisfies
〈J(z, v), J(z, v′)〉V = 〈z, z〉Z〈v, v′〉V .
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 12/27
Pseudo H-type algebras
All pseudo H-type algebras arises from the Cliffordalgebras Cl(Z, 〈. , 〉Z): If there is a representation
J : Z → End(V ) J2(z, ·) = −〈z, z〉Z IdV
such that V admits a scalar product 〈. , 〉V satisfying
〈J(z, v), v′〉V = −〈v, J(z, v′)〉V
then
n =(
V ⊕⊥ Z, [. , .], 〈. , 〉V + 〈. , 〉Z
)
is the pseudo H-type Lie algebra with the Lie bracket
〈J(z, v), v′〉V = 〈z, [v, v′]〉Z
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 13/27
Admissible Clifford module
When a Clifford Cl(Z, 〈· , ·〉Z )-module V
J2(z, ·) = −〈z, z〉Z IdV
admits a scalar product 〈· , ·〉V such that
〈J(z, v), v′〉V = −〈v, J(z, v′)〉V ?
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 14/27
Admissible Clifford module
When a Clifford Cl(Z, 〈· , ·〉Z )-module V
J2(z, ·) = −〈z, z〉Z IdV
admits a scalar product 〈· , ·〉V such that
〈J(z, v), v′〉V = −〈v, J(z, v′)〉V ?
Always! if 〈· , ·〉Z is positive definite
with 〈· , ·〉V positive definite =⇒
Classical H-type algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 14/27
Existence of adm. module
Given a Cl(Z, 〈· , ·〉Z )-module V , then V or V ⊕ V canbe equipped with a scalar product satisfying
(2) 〈J(z, v), v′〉V = −〈v, J(z, v′)〉V , for all z ∈ Z
P. Ciatti, 2000. Moreover
(V, 〈· , ·〉V ) or (V ⊕ V, 〈· , ·〉V ⊕V )
is a neutral space
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 15/27
List of pseudo H-type algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 16/27
Classical H-type algebras
For classical H-type algebras (n = V ⊕⊥ Z, [· , ·], (· , ·))there is a basis V = span{vα} and Z = span{zj} suchthat
[vα, vβ] =∑
j
Cjαβ
zj , Cjαβ
∈ Z.
G. Crandall, J. Dodziuk, Integral structures on H-type Lie algebras, J. Lie Theory 12 (2002), no.
1, 69-79.
P. Eberlein, Geometry of 2-step nilpotent Lie groups, Modern dynamical systems and applications,
Cambridge Univ. Press, Cambridge (2004), 67–101.
A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 39, 1951; Izv.
Akad. Nauk USSR, Ser. Mat. 13 (1949), 9-32.
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 17/27
General H-type algebras
Do the general H-type algebras
(N = V ⊕⊥ Z, [· , ·], 〈· , ·〉)
admit integer constants?
[vα, vβ] =∑
j
Cjαβ
zj , Cjαβ
∈ Z.
Answer is YES!
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 18/27
Atiyah-Bott periodicity of Clifford
algebras
Clr+8,s ∼ Clr,s⊗Cl8,0 ∼ Clr,s⊗R(16)
Clr,s+8 ∼ Clr,s⊗Cl0,8 ∼ Clr,s⊗R(16)
Clr+4,s+4 ∼ Clr,s⊗Cl4,4 ∼ Clr,s⊗R(16)
Clr,s+1 ∼ Cls,r+1
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 19/27
Symmetries of Clifford algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 20/27
Main idea of the proof
[vα, vβ] =∑
j
Cjαβ
zj , vα ∈ V, z ∈ Z.
J : Z → End(V )
Jzjvα =∑
β
Bjαβ
vβ.
〈Jzjvα, vβ〉V = 〈[vα, vβ ], zj〉Z
Cjαβ
= Bjαβ
νZj νVβ
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 21/27
Main idea of the proof
Given an orthonormal basis of Z, we construct anorthonormal basis for V
a. 〈w,w〉V =1
b. {w, Jziw, JziJzjw, JziJzjJzlw, JziJzjJzlJzmw},
1 ≤ i < j < l < m ≤ dimV is an o.n. basis
c. Jzi permute the basis for all i = 1, . . . , dimZ
using the Bott periodicity and some of symmetries ofClifford algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 22/27
List of pseudo H-type algebras
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 23/27
Isomorphism of pseudo H-type Lie
algebras
Cl1,0 ∼= C −→ n1,0 = H1 ∼ C⊕R
〈z, z〉Z = 1, v1, v2 = Jz(v1), Jz(v2) = −v1
〈v1, v1〉V = 1, 〈v2, v2〉V = 〈Jz(v1), Jz(v1)〉V = 1
[row , col.] v1 v2
v1 0 z
v2 −z 0
Cl0,1 ∼= R −→ n0,1 ∼ R2 ⊕R
〈z, z〉Z = −1, v1, v2 = Jz(v1), Jz(v2) = v1
〈v1, v1〉V = 1, 〈v2, v2〉V = 〈Jz(v1), Jz(v1)〉V = −1
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 24/27
Isomorphism of pseudo H-type Lie
algebras
n2,0 is isomorphic to n0,2 BUT NOT isomorphic to n1,1
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 25/27
Isomorphism of pseudo H-type Lie
algebras
THEOREM Autenried, Furutani, M
Two pseudo H-type Lie algebras nr,s and nt,u can beisomorphic only if
(r, s) = (t, u) or (r, s) = (u, t)
For example nr,0∼= n0,r, for r = 1, 2, 4, 8 mod 8
nr,8s∼= n8s,r, nr+4s,4s
∼= n4s,r+4s
n1,8∼= n8,1, n5,4
∼= n4,5
n2,3 6∼= n3,2
Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 26/27
The end
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