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 H 1 = span { X, Y } ⊕ span { Z } = V ⊕ Z , [ X, Y ] = Z is unique non vanishing commutator [ X, Y ] = Z Let ( · , · ) be an inner product such that X, Y, Z are orthonormal. Define J : Z × V → V an operator ( J ( z, v ) , u ) V := ( z, [ v, u ]) Z = ( z, ad v 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, ad v 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 ker(ad v ) ⊥ , ( · , · ) V ad v : � � ↔ ( Z , ( · , · ) Z ) ker(ad v ) ⊥ , ( · , · ) ker(ad v ) ⊥ � J ( · , v ): ( Z , ( · , · ) Z ) ↔ � Pseudo H -type algebras, integer structure constants and isomorphisms. – p. 3/27
Properties of J The operators ad v : � ker(ad v ) ⊥ , ( · , · ) ker(ad v ) ⊥ � ↔ ( Z , ( · , · ) Z ) J ( · , v ): ( Z , ( · , · ) Z ) ↔ � ker(ad v ) ⊥ , ( · , · ) ker(ad v ) ⊥ � 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, ad v 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, ad v v ′ ) Z is an isometry for any unit length vector v ∈ V . An H-type algebra n exists iff J 2 ( z, · ) = − ( z, z ) Z Id V . 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) � J 2 ( z, v ) , v ′ � V = −� J ( z, v ) , J ( z, v ′ ) � V = −� z, z � Z � v, v ′ � V � ( −� z, z � Z ) v, v ′ � V ⇒ = J 2 ( z, v ) = −� z, z � Z v J 2 ( z, · ) = −� z, z � Z Id V or (3) ⇒ (1) + (2) = (3) Pseudo H -type algebras, integer structure constants and isomorphisms. – p. 6/27
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) J 2 ( z, · ) = −� z, z � Z Id V (3) 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, 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 associative algebra 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 ( w 1 , . . . , w n ) is an orthonormal basis of W w k ⊗ w k = −� w k , w k � W I , w k ⊗ w l = − w l ⊗ w k , k � = 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 �→ J ( w, · ): V → V J ◦ J = J 2 ( w, · ): V → V w ⊗ w �→ −� w, w � W I �→ −� w, w � W Id V J 2 ( w, · ) = −� w, w � W Id V = ⇒ 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 � 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) J 2 ( z, · ) = −� z, z � Z Id V (3) 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 � 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) J 2 ( z, · ) = −� z, z � Z Id V (3) 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, ad v 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 Clifford algebras Cl( Z , � . , � Z ) : If there is a representation J 2 ( z, · ) = −� z, z � Z Id V J : Z → End( V ) 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 J 2 ( z, · ) = −� z, z � Z Id V 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 J 2 ( z, · ) = −� z, z � Z Id V 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 � J ( z, v ) , v ′ � V = −� v, J ( z, v ′ ) � V , for all z ∈ Z (2) P . Ciatti, 2000. Moreover or ( V, �· , ·� V ) ( 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 { v α } and Z = span { z j } such that � C j C j [ v α , v β ] = αβ ∈ Z . αβ z j , j 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? � C j C j [ v α , v β ] = αβ ∈ Z . αβ z j , j Answer is YES! Pseudo H -type algebras, integer structure constants and isomorphisms. – p. 18/27
Atiyah-Bott periodicity of Clifford algebras Cl r +8 ,s ∼ Cl r,s ⊗ Cl 8 , 0 ∼ Cl r,s ⊗ R (16) Cl r,s +8 ∼ Cl r,s ⊗ Cl 0 , 8 ∼ Cl r,s ⊗ R (16) Cl r +4 ,s +4 ∼ Cl r,s ⊗ Cl 4 , 4 ∼ Cl r,s ⊗ R (16) Cl r,s +1 ∼ Cl s,r +1 Pseudo H -type algebras, integer structure constants and isomorphisms. – p. 19/27
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