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 .

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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)⊥

)

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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

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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 .

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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 .

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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)

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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.

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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

)

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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

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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

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List of Clifford algebras

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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)

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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 .

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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

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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

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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

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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.

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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

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Symmetries of Clifford algebras

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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

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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

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List of pseudo H-type algebras

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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

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Isomorphism of pseudo H-type Lie

algebras

n2,0 is isomorphic to n0,2 BUT NOT isomorphic to n1,1

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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

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The end

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