Pseudo H -type algebras, integer structure constants and - - PowerPoint PPT Presentation

Pseudo H-type algebras, integer structure constants 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 are

• rthonormal.

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−1

v (·), 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

• r

(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 .

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, zZv, 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, zZv, v′V (−z, zZ)v, v′V = ⇒ J2(z, v) = −z, zZv

• r

J2(z, ·) = −z, zZ IdV

(3)

(1) + (2) = ⇒ (3)

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Relation to Clifford algebras

(1) J(z, v), J(z, v′)V = z, zZv, v′V . (2) J(z, v), v′V = −v, J(z, v′)V (3) J2(z, ·) = −z, zZ IdV

The first property is the composition of quadratic forms and The last property is defining property for Clifford algebra.

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 associative algebra with unit I, product ⊗, factorized by the relation

w ⊗ w = −w, wWI

• r
• w ⊗ u + u ⊗ w = −2w, uWI
• 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 associative algebra with unit I, product ⊗, factorized by the relation

w ⊗ w = −w, wWI

• r
• w ⊗ u + u ⊗ w = −2w, uWI
• If (w1, . . . , wn) is an orthonormal basis of W

wk ⊗ wk = −wk, wkWI, wk ⊗ wl = −wl ⊗ wk, k = 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 → J(w, ·): V → V w ⊗ w → J ◦ J = J2(w, ·): V → V −w, wWI → −w, wW IdV = ⇒ J2(w, ·) = −w, wW 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, zZv, v′V . (2) J(z, v), v′V = −v, J(z, v′)V (3) J2(z, ·) = −z, zZ IdV

Question: given (3) can we construct a general H-type algebra?

(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, zZv, v′V . (2) J(z, v), v′V = −v, J(z, v′)V (3) J2(z, ·) = −z, zZ IdV

Question: given (3) can we construct a general H-type algebra?

(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 Lie algebra such that

(Z, · , ·Z), (V, · , ·V ) are non degenerate

The Lie algebra N is a called pseudo H-type Lie algebra 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, zZv, v′V .

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Pseudo H-type algebras

All pseudo H-type algebras arises from the Clifford algebras Cl(Z, . , Z): If there is a representation

J : Z → End(V ) J2(z, ·) = −z, zZ 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, zZ 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, zZ 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 can be 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 )

• r

(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} such that

[vα, vβ] =

• j

Cj

αβzj,

Cj

αβ ∈ 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. Mal´

cev, 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

Cj

αβzj,

Cj

αβ ∈ 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

Pseudo H-type algebras, integer structure constants and isomorphisms. – p. 20/27

Main idea of the proof

[vα, vβ] =

• j

Cj

αβzj,

vα ∈ V, z ∈ Z. J : Z → End(V ) Jzjvα =

• β

Bj

αβvβ.

Jzjvα, vβV = [vα, vβ], zjZ Cj

αβ = Bj αβνZ j νV β

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 an

• rthonormal basis for V
• a. w, wV =1
• b. {w, Jziw, JziJzjw, JziJzjJzlw, JziJzjJzlJzmw},

1 ≤ i < j < l < m ≤ dim V is an o.n. basis

• c. Jzi permute the basis for all i = 1, . . . , dim Z

using the Bott periodicity and some of symmetries of Clifford 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, zZ = 1, v1, v2 = Jz(v1), Jz(v2) = −v1 v1, v1V = 1, v2, v2V = Jz(v1), Jz(v1)V = 1 [row , col.] v1 v2 v1 z v2 −z Cl0,1 ∼ = R − →

n0,1 ∼ R2 ⊕ R

z, zZ = −1, v1, v2 = Jz(v1), Jz(v2) = v1 v1, v1V = 1, v2, v2V = 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 be isomorphic only if

(r, s) = (t, u)

• r

(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 ∼

= n3,2

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The end

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