On para-Kähler and hyper-para-Kähler Lie algebras
aa r X i v : . [ m a t h . DG ] D ec On para-K¨ahler and hyper-para-K¨ahler Lie algebras
Sa¨ıd Benayadi, Mohamed Boucetta a,b a Universit´e de Lorraine, Laboratoire IECL, CNRS-UMR 7502,Ile du Saulcy, F-57045 Metz cedex 1, France.e-mail: [email protected] b Universit´e Cadi-AyyadFacult´e des sciences et techniquesBP 549 Marrakech Maroce-mail: [email protected]
Abstract
We study Lie algebras admitting para-K¨ahler and hyper-para-K¨ahler structures. We give new characterizations ofthese Lie algebras and we develop many methods to build large classes of examples. Bai considered para-K¨ahlerLie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain somenew results. The study of para-K¨ahler and hyper-para-K¨ahler is intimately linked to the study of left symmetricalgebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of leftsymmetric algebras and a complete description of all associative algebras admitting an invariant symplectic form. Wegive also all four dimensional hyper-para-K¨ahler Lie algebras.
Keywords: para-K¨ahler Lie algebra, hyper-para-K¨ahler Lie algebra, symplectic Lie algebras, left symmetricalgebras, S -matrix
1. Introduction A para-complex structure on a 2 n -dimensional manifold M is a field K of involutive endomorphisms ( K = Id T M )such that the eigendistributions T ± M with eigenvalues ± n and are integrable. In the presenceof a pseudo-Riemannian metric this notion leads to the notion of para-K¨ahler manifolds. A para-K¨ahler structureon a manifold M is a pair ( g , K ) where g is a pseudo-Riemannian metric and K is a parallel skew-symmetric para-complex structure. If ( g , K ) is a para-K¨ahler structure on M , then ω = g ◦ K is a symplectic structure and the ± T ± M of K are two integrable ω -Lagrangian distributions. Due to this, a para-K¨ahler structurecan be identified with a bi-Lagrangian structure ( ω, T ± M ) where ω is a symplectic structure and T ± M are two inte-grable Lagrangian distributions. If ( M , g , K ) is a para-K¨ahler manifold and J is a parallel field of skew-symmetricendomorphisms such that J = − Id T M and JK = − K J then ( M , g , K , J ) is called a hyper-para-K¨ahler manifold or hyper-symplectic manifold. The notion of almost para-complex structure (or almost product structure) on a manifoldwas introduced by P.K. Rasevskii [19] and P. Libermann [17]. The paper [11] contains a survey on para-K¨ahler ge-ometries. Hyper-para-K¨ahler structures were introduced by N. Hitchin in [14] and have become an important subjectof study lately, due mainly to their applications in theoretical physics (specially in dimension 4). See for instance [8],where there is a discussion on the relationship between hyper-para-K¨ahler metrics and the N = G , the metric and the para-complex structure are considered left-invariant, they are bothdetermined by their restrictions to the Lie algebra g of G . In a such situation, ( g , g e , K e ) is called para-K¨ahler Lie This research was conducted within the framework of Action concert´ee CNRST-CNRS Project SPM04 / Preprint submitted to Elsevier November 12, 2018 lgebra . We recover also the notion of hyper-para-K¨ahler Lie algebra when we start from a left invariant hyper-para-K¨ahler structure on a Lie group. Para-K¨ahler and hyper-para-K¨ahler Lie algebras has been studied by many authors[1, 2, 4, 5, 6].This paper is devoted to the study of para-K¨ahler and hyper-para-K¨ahler Lie algebras. We present some known resultsby adopting a new approach which we think simplify both the presentation and the proofs. In the large part of thepaper, we give some new results which permit a better understanding of this algebras and the construction of a richclasses of non trivial new examples. The basic tools of the study of para-K¨ahler and hyper-para-K¨ahler Lie algebrasare two types of algebras: left symmetric algebras which have been studied by many authors and a less known class,namely, left symmetric algebras endowed with invariant symplectic forms called in [3] special symplectic Lie alge-bras. We call such algebras symplectic left symmetric algebras. Our study leads incidentally to the construction of alarge classes of left symmetric algebras, symplectic left symmetric algebras and symplectic Lie algebras.The paper is organized as follows. In Sections 2 and 3 we recall some basic definitions and we present, by using anew approach, some known characterizations of para-K¨ahler Lie algebras. We give a particular attention to the notionof left symmetric bialgebras introduced by Bai [4]. We introduce the notion of quasi S -matrices as a generalizationof S -matrices introduced by Bai. Proposition 3.7 describing the Lie algebra structure of the para-K¨ahler Lie algebraassociated to a quasi S -matrix will play a crucial role in Sections 6-7. It shows also (see Remark 2 ( b )) that a quasi S -matrix on a left symmetric algebra U defines a Lie triple system on U ∗ (see [15, 18, 20] for the definition andproperties of Lie triple systems). In Section 4, we develop some general methods to build new examples of para-K¨ahler Lie algebras. In Section 5, we give a new characterization of hyper-para-K¨ahler Lie algebras based on a notionof compatibility between two left symmetric products on a given vector space (see Theorem 5.1 and Definition 5.1).Sections 6-7 are devoted to the study of quasi S -matrices on symplectic Lie algebras, on symplectic left symmetricalgebras and on pseudo-Riemannian flat Lie algebras. On a symplectic Lie algebra with its canonical left symmetricproduct the set of quasi S -matrices is in bijection with the set of solutions of an equation which generalizes themodified Yang-Baxter equation (see Proposition 6.1). As a consequence, we find a method to build a new class ofpara-K¨ahler Lie algebras (see Theorem 6.1) and actually a new class of Lie algebras with a Lie triple system (seeRemark 4). On a symplectic left symmetric algebra or a pseudo-Riemannian flat Lie algebra, the set of S -matricesis in bijection with the set of operators generalizing O -operators (see Proposition 7.1). As a consequence, we finda method to build a new class of para-K¨ahler and hyper-para-K¨ahler Lie algebras (see Theorems 7.1-7.2). We getalso a new class of Lie algebras with a Lie triple system (see Remark 5). In Section 8 we give all four dimensionalpara-K¨ahler Lie algebras. We use a method which is di ff erent from the one used in [1] and which has the advantageof simplifying enormously the calculations. We devote Section 9 to a complete description of associative symplecticleft symmetric algebras (see Theorems 9.1-9.2). Notations.
For a Lie algebra g , its bracket will be denoted by [ , ] and for any u ∈ g , ad u is the endomorphism of g given by ad u ( v ) = [ u , v ]. If A : g −→ g is an endomorphism, the Nijenhuis torsion of A is given by N A ( u , v ) : = [ Au , Av ] − A [ Au , v ] − A [ u , Av ] + A [ u , v ] . (1)If ( U , . ) is an algebra, for any u ∈ U , L u , R u : U −→ U denote the left and the right multiplication by u given byL u ( v ) = u . v and R u ( v ) = v . u . The commutator of ( U , . ) is the bracket on U given by [ u , v ] = u . v − v . u . The curvatureof ( U , . ) is the tensor K given by K( u , v ) : = [L u , L v ] − L [ u , v ] . Then, for any u , v , w ∈ U , we have the Bianchi identity I [ u , [ v , w ]] = I K( u , v ) w , (2)where H stands of the cyclic sum. The product on U is called Lie-admissible if its commutator is a Lie bracket, i.e.,for any u , v , w ∈ U , I [ u , [ v , w ]] = I K( u , v ) w = . V be a vector space and F is an endomorphism of V . We denote by F t : V ∗ −→ V ∗ the dual endomorphism. Forany X ∈ V and α ∈ V ∗ we denote α ( X ) by ≺ α, X ≻ . The phase space of V is the vector space Φ ( V ) : = V ⊕ V ∗ endowedwith the two nondegenerate bilinear forms h , i and Ω given by h u + α, v + β i = ≺ α, v ≻ + ≺ β, u ≻ and Ω ( u + α, v + β ) = ≺ β, u ≻ − ≺ α, v ≻ . We denote by K : Φ ( V ) −→ Φ ( V ) the endomorphism given by K ( u + α ) = u − α .Let ω ∈ ∧ V ∗ which is nondegenerate. We denote by ♭ : V −→ V ∗ the isomorphism given by ♭ ( v ) = ω ( v , . ). Put T ( V ) : = V × V and define h , i , Ω , K , J on T ( V ) by Ω [( u , v ) , ( w , z )] = ω ( z , u ) − ω ( v , w ) , h ( u , v ) , ( w , z ) i = ω ( z , u ) + ω ( v , w ) , K ( u , v ) = ( u , − v ) and J ( u , v ) = ( − v , u ) . Finally, if ρ : g −→ End( V ) is a representation of a Lie algebra, we denote by ρ ∗ : g −→ End( V ∗ ) the dual representa-tion given by ρ ∗ ( X )( α ) = − ρ ( X ) t ( α ).
2. Some definitions
In this section, we recall the definitions of di ff erent types of algebraic structures used through this paper. • A complex structure on a Lie algebra g is an isomorphism J : g −→ g satisfying J = − Id g and N J =
0. Acomplex structure J is called abelian if, for any u , v ∈ g ,[ Ju , Jv ] = [ u , v ] . A para-complex structure on a Lie algebra g is an isomorphism K : g −→ g satisfying K = Id G , N K = K + Id g ) = dim ker( K − Id g ). A para-complex structure K is called abelian if, for any u , v ∈ g ,[ Ku , Kv ] = − [ u , v ] . A complex product structure on g is a couple ( J , K ) where J is a complex structure, K is a para-complex structureand K J = − JK . • A pseudo-Riemannian Lie algebra is a finite dimensional Lie algebra ( g , [ , ]) endowed with a bilinear sym-metric nondegenerate form h , i . The associated Levi-Civita product is the product on g , ( u , v ) u . v , given byKoszul’s formula 2 h u . v , w i = h [ u , v ] , w i + h [ w , u ] , v i + h [ w , v ] , u i . (3)This product is entirely determined by the facts that it is Lie-admissible, i.e., [ u , v ] = u . v − v . u and, for any u ∈ g ,the left multiplication by u is skew-symmetric with respect to h , i . We call ( g , [ , ]) flat when its Levi-Civitaproduct has a vanishing curvature. • An algebra ( U , . ) is called left symmetric ifass( u , v , w ) = ass( v , u , w ) , where ass( u , v , w ) = ( u . v ) . w − u . ( v . w ). This relation is equivalent to the vanishing of the curvature of ( U , . ). Therelation (2) implies that a left symmetric product is Lie-admissible. If ( U , . ) is a left symmetric algebra then theLie algebra ( U , [ , ]) has two representations, namely, ad U : U −→ End( U ) , u ad u and L U : U −→ End( U ) , u L u .Any associative algebra is a left symmetric algebra and if a left symmetric algebra is abelian then it is associa-tive. 3 A symplectic Lie algebra is a Lie algebra ( g , ω ) endowed with a bilinear skew-symmetric nondegenerate form ω such that, for any u , v , w ∈ g , ω ([ u , v ] , w ) + ω ([ v , w ] , u ) + ω ([ w , u ] , v ) = . It is a well-known result [12] that the product a : g × g −→ g given by ω ( a ( u , v ) , w ) = − ω ( v , [ u , w ]) (4)induces on g a left symmetric algebra structure satisfying a ( u , v ) − a ( v , u ) = [ u , v ]. We call a the left symmetricproduct associated to ( g , ω ). • A symplectic left symmetric algebra is a left symmetric algebra ( U , . ) endowed with a bilinear skew-symmetricnondegenerate form ω which is invariant, i.e., for any u , v , w ∈ U , ω ( u . v , w ) + ω ( v , u . w ) = . This implies that ( U , [ , ] , ω ) is a symplectic Lie algebra. Symplectic left symmetric algebras, called in [5]special symplectic Lie algebras, play a central role in the study of hyper-para-K¨ahler Lie algebras (see Section5).
3. Para-K¨ahler Lie algebras
The notion of para-K¨ahler Lie algebra is very subtle and has many equivalent definitions depending on if oneemphasizes on its pseudo-Riemannian metric and the associated Levi-Civita product or on its symplectic form and itsassociated left symmetric product. It has also many characterizations. There is at least three such characterizationsin [4]: a para-K¨ahler Lie algebra can be characterized as the phase space of a Lie algebra, as a matched pair ofLie algebras or as a left symmetric bialgebra. In this section, we choose the pseudo-Riemannian point of view andwe give a new characterization based on Bianchi identity (2). This characterization has the advantage of leadingeasily to the notions of left symmetric bialgebra and S -matrix introduced in [4]. To end this section, we introduce ageneralization of the notion of S -matrix and we give a precise description of the para-K¨ahler Lie algebras associatedto these generalized S -matrices.A para-K¨ahler Lie algebra is a pseudo-Riemannian Lie algebra ( g , h , i ) endowed with an isomorphism K : g −→ g satisfying K = Id g , K is skew-symmetric with respect to h , i and K is invariant with respect to the Levi-Civitaproduct, i.e., L u ◦ K = K ◦ L u , for any u ∈ g . A para-K¨ahler Lie algebra ( g , h , i , K ) carries a natural bilinearskew-symmetric nondegenerate form Ω defined by Ω ( u , v ) = h Ku , v i , and one can see easily that:1. ( g , K ) is a para-complex Lie algebra,2. ( g , Ω ) is a symplectic Lie algebra,3. g = g ⊕ g − where g = ker( K − Id g ) and g − = ker( K + Id g ),4. g and g − are subalgebras isotropic with respect to h , i and Lagrangian with respect to Ω ,5. for any u ∈ g , u . g ⊂ g and u . g − ⊂ g − (the dot is the Levi-Civita product).A para-K¨ahler Lie algebra carries two products: the Levi-Civita product and the left symmetric product a associatedto ( g , Ω ). The following proposition clarifies the relations between them. Its proof is an easy computation. Proposition 3.1.
Let ( g , h , i , Ω , K ) be a para-K¨ahler Lie algebra. Then, for any u , v ∈ g and α, β ∈ g − ,u . v = a ( u , v ) and α.β = a ( α, β ) . In particular, g and g − are left symmetric algebras. Let ( g , h , i , Ω , K ) be a para-K¨ahler Lie algebra. For any u ∈ g − , let u ∗ denote the element of ( g ) ∗ givenby ≺ u ∗ , v ≻ = h u , v i . The map u u ∗ realizes an isomorphism between g − and ( g ) ∗ . Thus we can identify( g , h , i , Ω , K ) to the phase space ( Φ ( g ) , h , i , Ω , K ). With this identification in mind and according to Proposition4.1, the Levi-Civita product induces a product on g and ( g ) ∗ which coincides with the a ffi ne product a . Thus both g and ( g ) ∗ carry a left symmetric algebra structure. For any u ∈ g and for any α ∈ ( g ) ∗ , we denote by L u : g −→ g and L α : ( g ) ∗ −→ ( g ) ∗ the left multiplication by u and α , respectively, i.e., for any v ∈ g and any β ∈ ( g ) ∗ ,L u v = u . v = a ( u , v ) and L α β = α.β = a ( α, β ) . The right multiplications R u and R α are defined in a similar manner. The following proposition shows that the Levi-Civita product and the a ffi ne product on g identified with Φ ( g ) are entirely determined by their restrictions to g and( g ) ∗ . The proof of this proposition is straightforward. Proposition 3.2.
Let g be a para-K¨ahler Lie algebra identified with Φ ( g ) as above. Then For any u ∈ g and for any α ∈ ( g ) ∗ , u .α = − L tu α and α. u = − L t α u . For any u ∈ g and for any α ∈ ( g ) ∗ , a ( u , α ) = R t α u − ad tu α and a ( α, u ) = − ad t α u + R tu α, where ad u : g −→ g and ad α : ( g ) ∗ −→ ( g ) ∗ are given by ad u v = [ u , v ] and ad α β = [ α, β ] . Conversely, let U be a finite dimensional vector space and U ∗ its dual space. We suppose that both U and U ∗ havea structure of left symmetric algebra. We extend the products on U and U ∗ to Φ ( U ) by putting, for any X , Y ∈ U andfor any α, β ∈ U ∗ , ( X + α ) . ( Y + β ) = X . Y − L t α Y − L tX β + α.β. (5)We consider the two bilinear maps ρ : U × U ∗ −→ End( U ) and ρ ∗ : U ∗ × U −→ End( U ∗ ) defined by ρ ( X , α ) = [L X , L t α ] + L L t α X + L t L tX α and ρ ∗ ( α, X ) = [L α , L tX ] + L L tX α + L t L t α X . (6)Note that the endomorphism ρ ∗ ( α, X ) is the dual of ρ ( X , α ). We give now a new characterization of para-K¨ahler Liealgebras using Bianchi identity. Proposition 3.3.
With the hypothesis above, the product on Φ ( U ) given by (5) is Lie-admissible if and only if ρ ( X , α ) Y = ρ ( Y , α ) X and ρ ∗ ( α, X ) β = ρ ∗ ( β, X ) α (7) for any X , Y ∈ U and any α, β ∈ U ∗ . Moreover, this product is left symmetric if and only if ρ ( X , α ) = , for any X ∈ Uand any α ∈ U ∗ . Proof.
According to Bianchi identity (2), the product given by (5) is Lie-admissible if and only if, for any u , v , w ∈ Φ ( U ), K( u , v ) w + K( v , w ) u + K( w , u ) v = , where K is the curvature of the product. Since the products on U and U ∗ are left symmetric, for any X , Y , Z ∈ U andfor any α, β, γ ∈ U ∗ , K( X , Y ) Z = K( α, β ) γ = . Thus the Bianchi identity is equivalent toK( X , Y ) α + K( Y , α ) X + K( α, X ) Y = , ( ∗ )K( X , α ) β + K( α, β ) X + K( β, X ) α = , ( ∗∗ )for any X , Y ∈ U and for any α, β ∈ U ∗ . Now, we have obviously,K( X , Y ) α = (cid:0) [L X , L Y ] − L [ X , Y ] (cid:1) t α = . On the other hand, a direct computation givesK( Y , α ) X = ρ ( Y , α ) X and K( α, X ) Y = − ρ ( X , α ) Y . So ( ∗ ) is equivalent to ρ ( X , α ) Y = ρ ( Y , α ) X . A similar computation shows that ( ∗∗ ) is equivalent to ρ ∗ ( α, X ) β = ρ ∗ ( β, X ) α . The second part of the proposition follows easily from what above. (cid:3) efinition 3.1. Two left symmetric products on U and U ∗ satisfying (7) will be called Lie-extendible . Thus we get the following result.
Theorem 3.1.
Let ( U , . ) and ( U ∗ , . ) two Lie-extendible left symmetric products. Then ( Φ ( U ) , h , i , K ) endowed withthe Lie algebra bracket associated to the product given by (5) is a para-K¨ahler Lie algebra. Moreover, all para-K¨ahlerLie algebras are obtained in this way. Example 1.
Let ( U , . ) be a left symmetric algebra. Then the left symmetric product on U and the trivial left symmetricproduct on U ∗ are Lie-extendible so ( Φ ( U ) , h , i , K ) endowed with the Lie algebra bracket associated to the leftsymmetric product ( X + α ) ⊲ ( Y + β ) = X . Y − L tX β (8) is a para-K¨ahler Lie algebra. We denote by [ , ] ⊲ the Lie bracket associated to ⊲ . We have [ X + α, Y + β ] ⊲ = [ X , Y ] − L tX β + L tY α. This is just the semi-direct product of ( U , [ , ]) with U ∗ endowed with the trivial bracket and the action of U on U ∗ is given by L ∗ U . Moreover, it is easy to check that ( Φ ( U ) , [ , ] ⊲ , h , i ) is a flat pseudo-Riemannian Lie algebra and ( Φ ( U ) , ⊲, Ω ) is a symplectic left symmetric algebra (see also Proposition 4.3 [5]). In [4], Bai gave a characterization of para-K¨ahler Lie algebras similar to the one used for Lie bialgebras. He calledthese structures left symmetric bialgebras . Let us present this point of view in a new way by using Proposition 3.3.We consider two left symmetric algebras ( U , . ) and ( U ∗ , . ). The products on U and U ∗ define by duality, respectively,two maps µ : U ∗ −→ U ∗ ⊗ U ∗ and ξ : U −→ U ⊗ U . As Lie algebras U and U ∗ have two representations Ψ U : U −→ End( U ⊗ U ) and Ψ U ∗ : U ∗ −→ End( U ∗ ⊗ U ∗ ) given by Ψ U = L U ⊗ ad U and Ψ U ∗ = L U ∗ ⊗ ad U ∗ . A direct computation gives, for any X , Y ∈ U and for any α, β ∈ U ∗ , ≺ β, ρ ( X , α ) Y − ρ ( Y , α ) X ≻ = Ψ U ( X )( ξ ( Y ))( α, β ) − Ψ U ( Y )( ξ ( X ))( α, β ) − ξ ([ X , Y ])( α, β ) , ≺ ρ ∗ ( α, X ) β − ρ ∗ ( β, X ) α, Y ≻ = Ψ U ∗ ( α )( µ ( β ))( X , Y ) − Ψ U ∗ ( β )( µ ( α ))( X , Y ) − µ ([ α, β ])( X , Y ) . By using Proposition 3.3, we recover a result of Bai (see [4] Theorem 4.1).
Proposition 3.4.
The product on Φ ( U ) given by (5) is Lie-admissible if and only if ξ is a 1-cocycle of ( U , [ , ]) withrespect to the representation Ψ U and µ is a 1-cocycle of ( U ∗ , [ , ]) with respect to the representation Ψ U ∗ , i.e., for anyX , Y ∈ U , α, β ∈ U ∗ , ξ ([ X , Y ]) = Ψ U ( X )( ξ ( Y )) − Ψ U ( Y )( ξ ( X )) ,µ ([ α, β ]) = Ψ U ∗ ( α )( µ ( β )) − Ψ U ∗ ( β )( µ ( α )) . We consider now the case where ξ is a co-boundary. Indeed, let ( U , . ) be a left symmetric algebra and ξ : U −→ U ⊗ U a co-boundary of ( U , [ ]) with respect to Ψ U , i.e., ξ = δ r where r ∈ U ⊗ U . By duality, ξ define a product on U ∗ by ≺ α.β, X ≻ = r(L tX α, β ) + r( α, ad tX β ) = L X r( α, β ) − ≺ L t r ( α ) β, X ≻ , (9)where r : U ∗ −→ U is given by ≺ β, r ( α ) ≻ = r( α, β ) and L X r( α, β ) = r(L tX α, β ) + r( α, L tX β ) . According to Proposition 3.3, to get a para-K¨ahler Lie algebra structure on Φ ( U ), ( U ∗ , . ) must be a left symmetricalgebra and the couple ( U , . ) , ( U ∗ , . ) must be Lie-extendible. Note that, since ξ = δ r, the first equation in (7) holds.Let us find out under which conditions the second equation in (7) holds and ( U ∗ , . ) is a left symmetric algebra. Putr = a + s where a is skew-symmetric and s is symmetric and define L( a ) ∈ U ∗ ⊗ U ⊗ U byL( a )( X , α, β ) = L X a ( α, β ) .
6t follows immediately from (9) that, for any α, β ∈ U ∗ and X ∈ U ,L t α X = r ◦ L tX α + [ X , r ( α )] and ≺ [ α, β ] , X ≻ = ≺ L t r ( β ) α − L t r ( α ) β, X ≻ + X a ( α, β ) . (10)We consider the two representations Q : U −→ End( U ⊗ U ⊗ U ) and P : U −→ End( U ∗ ⊗ U ⊗ U ) given byQ = L U ⊗ L U ⊗ ad U and P = L ∗ U ⊗ L U ⊗ L U . We define also ∆ (r) ∈ U ⊗ U ⊗ U ≃ End( U ∗ ⊗ U ∗ , U ) by ∆ (r)( α, β ) = r ([ α, β ]) − [r ( α ) , r ( β )] . (11) Proposition 3.5.
For any X , Y ∈ U and α, β, γ ∈ U ∗ , we have ≺ ρ ∗ ( α, X ) β − ρ ∗ ( β, X ) α, Y ≻ = P ( X )(L( a ))( Y , α, β ) , ass( α, β, γ ) − ass( β, α, γ ) = ≺ γ, − Q( X )( ∆ (r))( α, β ) + r ( ρ ∗ ( α, X ) β − ρ ∗ ( β, X ) α ) ≻ . Proof.
Let us compute first the associator of α, β, γ ∈ U ∗ with respect to the product given by (9). We have, forany X ∈ U , ≺ ass( α, β, γ ) , X ≻ = ≺ α. ( β.γ ) , X ≻ − ≺ ( α.β ) .γ, X ≻ = r(L tX α, β.γ ) + r( α, ad tX ( β.γ )) − r(L tX ( α.β ) , γ ) − r( α.β, ad tX γ ) = ≺ β.γ, r (L tX α ) ≻ + ≺ ad tX ( β.γ ) , r ( α ) ≻ − ≺ γ, r (L tX ( α.β )) ≻ − ≺ ad tX γ, r ( α.β ) ≻ = r(L t r (L tX α ) β, γ ) + r( β, ad t r (L tX α ) γ ) + r(L t [ X , r ( α )] β, γ ) + r( β, ad t [ X , r ( α )] γ ) − ≺ γ, r (L tX ( α.β )) ≻ − ≺ γ, [ X , r ( α.β )] ≻ = ≺ γ, r (cid:18) L t r (L tX α ) β (cid:19) ≻ + ≺ γ, [r (L tX α ) , r ( β )] ≻ + ≺ γ, r (cid:16) L t [ X , r ( α )] β (cid:17) ≻ + ≺ γ, [[ X , r ( α )] , r ( β )] ≻ − ≺ γ, r (L tX ( α.β )) ≻ − ≺ γ, [ X , r ( α.β )] ≻ . On the other hand, Q ( X )( ∆ (r))( α, β ) = [ X , ∆ (r)( α, β )] − ∆ (r)(L ∗ X α, β ) − ∆ (r)( α, L ∗ X β ) = [ X , r ([ α, β ])] − [ X , [r ( α ) , r ( β )]] + r ([L tX α, β ]) − [r (cid:16) L tX α (cid:17) , r ( β )] + r ([ α, L tX β ]) − [r ( α ) , r (cid:16) L tX β (cid:17) ] . So ass( α, β, γ ) − ass( β, α, γ ) + ≺ γ, Q ( X )( ∆ (r))( α, β ) ≻ = ≺ γ, r ( A ) ≻ , where A = L t r (L tX α ) β − L t r (L tX β ) α + L t [ X , r ( α )] β − L t [ X , r ( β )] α − L tX ([ α, β ]) + [L tX α, β ] + [ α, L tX β ] . By using the first relation in (10), we get A = = L t L t α X β − L t L t β X α − L tX ([ α, β ]) + [L tX α, β ] + [ α, L tX β ] . Now, according to (6), we have ρ ∗ ( α, X ) β = [L α , L tX ] β + L L tX α β + L t L t α X β = α. (L tX β ) − L tX ( α.β ) + (L tX α ) .β + L t L t α X β, so A = ρ ∗ ( α, X ) β − ρ ∗ ( β, X ) α and the second assertion follows. On the other hand, by using the second relation in(10), we get for any Y ∈ U , ≺ A , Y ≻ = − X . Y a ( α, β ) + Y a (L tX α, β ) + Y a ( α, L tX β ) , and the first assertion follows. (cid:3) So we get the following result. 7 heorem 3.2.
Let ( U , . ) be a left symmetric algebra and r = a + s ∈ U ⊗ U. Then the product given by (9) is leftsymmetric and the left symmetric products on ( U , U ∗ ) are Lie-extendible if and only if for any X ∈ U Q( X )( ∆ (r)) = and P( X )(L( a )) = . In this case, ( Φ ( U ) , h , i , Ω , K ) endowed with the Lie bracket associated to the product given by (5) is a para-K¨ahlerLie algebra. We have immediately the following corollary.
Corollary 3.1. If a is L U -invariant, i.e. L( a ) = , then the product given by (9) is left symmetric and the left symmetricproducts on ( U , U ∗ ) are Lie-extendible if and only if ∆ (r) is Q-invariant. Actually the statement of Theorem 3.2 is the same as the one of Theorem 5.4 in [4]. To show this, let us investigatethe relation between ∆ (r) and [[r , r]] appearing in Bai’s Theorem. Let ( U , . ) be a left symmetric algebra and r = P i a i ⊗ b i . In [4], Bai defines [[r , r]] by[[r , r]] = r . r − r . r + [r , r ] − [r , r ] − [r , r ] , where r . r = X i , j a i . a j ⊗ b j ⊗ b i , r . r = X i , j b j ⊗ a i . a j ⊗ b i , [r , r ] = X i , j a j ⊗ [ a i , b j ] ⊗ b i , [r , r ] = X i , j [ a i , b j ] ⊗ a j ⊗ b i , [r , r ] = X i , j a i ⊗ a j ⊗ [ b i , b j ] . Proposition 3.6.
For any α, β, γ ∈ U ∗ , we have [[r , r]]( α, β, γ ) = ≺ γ, ∆ (r)( α, β ) ≻ . Proof.
Recall that according to (10), for any X ∈ U , ≺ [ α, β ] , X ≻ = ≺ L t r ( β ) α − L t r ( α ) β, X ≻ + X a ( α, β ) , a = P i ( a i ⊗ b i − b i ⊗ a i ) is the skew-symmetric part of r. We haver ( α ) = X i ≺ α, a i ≻ b i and r ( β ) = X i ≺ β, a i ≻ b i . So − ≺ γ, [r ( α ) , r ( β )] ≻ = − X i , j ≺ α, a i ≻≺ β, a j ≻≺ γ, [ b i , b j ] ≻ = − [r , r ]( α, β, γ ) . Now r(L t r ( β ) α, γ ) = X j ≺ β, a j ≻ r(L tb j α, γ ) = X i , j ≺ β, a j ≻≺ L tb j α, a i ≻≺ γ, b i ≻ = X i , j ( b j . a i ) ⊗ a j ⊗ b i ( α, β, γ ) . In the same way, we get − r(L t r ( α ) β, γ ) = − X i , j a j ⊗ ( b j . a i ) ⊗ b i ( α, β, γ ) .
8n the other hand, 2r(L a ( α, β ) , γ ) = X i ≺ L a ( α, β ) , a i ≻≺ γ, b i ≻ = X i a (L ta i α, β ) ≺ γ, b i ≻ + X i a ( α, L ta i β ) ≺ γ, b i ≻ , X i a (L ta i α, β ) ≺ γ, b i ≻ = X i , j (cid:16) ≺ L ta i α, a j ≻≺ β, b j ≻ − ≺ L ta i α, b j ≻≺ β, a j ≻ (cid:17) ≺ γ, b i ≻ = X i , j (cid:16) ( a i . a j ) ⊗ b j ⊗ b i − ( a i . b j ) ⊗ a j ⊗ b i (cid:17) ( α, β, γ ) , = r . r ( α, β, γ ) − X i , j ( a i . b j ) ⊗ a j ⊗ b i ( α, β, γ ) , X i a ( α, L ta i β ) ≺ γ, b i ≻ = X i , j (cid:16) a j ⊗ ( a i . b j ) ⊗ b i − b j ⊗ ( a i . a j ) ⊗ b i (cid:17) ( α, β, γ ) = X i , j a j ⊗ ( a i . b j ) ⊗ b i ( α, β, γ ) − r . r ( α, β, γ ) . By combining all what above we get the desired formula. (cid:3)
Remark 1. This proposition shows that statement of Theorem 3.2 is the same as the one of Theorem 5.4 in[4]. However, our proof is more easier because the expression of ∆ (r) is more simple to handle than the one of [[r , r]] . The practical nature of ∆ (r) will be crucial later, in particular, in Sections 6-7. Let ( U , . ) be a left symmetric algebra and r ∈ U ⊗ U. According to Proposition 3.6, [[ r , r ]] = i ff r is a Liealgebra endomorphism. This generalizes Theorem 6.6 in [4], stated in the case when r is symmetric.Suppose now that r is symmetric and r is an isomorphism. One can see easily by using (10) that, for anyX , Y , Z ∈ U, ≺ r − ( Z ) , ∆ (r)(r − ( X ) , r − ( Y )) = B ( X , Y . Z ) − B ( Y , X . Z ) − B ( Z , [ X , Y ]) , where B ∈ U ∗ ⊗ U ∗ is given by B ( X , Y ) = ≺ r − ( X ) , Y ≻ . So [[ r , r ]] = i ff B is 2-cocycle of ( U , . ) . This prove ina di ff erent way Theorem 6.3 in [4]. Let introduce now a key notion in our work, namely, the notion of quasi S -matrix as a generalization of the one of S -matrix appeared first in [4].Let U be a left symmetric algebra. A quasi S -matrix of U is a r ∈ U ⊗ U such that its skew-symmetric part isL U -invariant and [[r , r]] is Q -invariant. Recall that a S -matrix of U is a r ∈ U ⊗ U which is symmetric and satisfying[[ r , r ]] = . (12)In what follows, we focus our attention on the Lie algebra structure on Φ ( U ) associated to a quasi S -matrix. We showthat Lie algebra can be described in a precise and simple way. Indeed, let r be a quasi S -matrix. Then, accordingto Theorem 3.2, the product on U ∗ given by (9) is left symmetric and ( Φ ( U ) , [ , ] r , h , i , K ) is a para-K¨ahler Liealgebra, where [ X + α, Y + β ] r = [ X , Y ] − L tX β − L t α Y + L tY α + L t β X + [ α, β ] . We have shown in Example 1 that Φ ( U ) carries a left symmetric product ⊲ and its associated Lie bracket [ , ] ⊲ induceson Φ ( U ) a para-K¨ahler Lie algebra structure. We define a new bracket on Φ ( U ) by putting[ X + α, Y + β ] ⊲, r = [ X + α, Y + β ] ⊲ + ∆ (r)( α, β ) . (13)The following proposition has been inspired to us by a result appeared in [13] in the context of Lie bialgebras and R -matrices (see Proposition 4.2.1.1 of [13]). Proposition 3.7. ( Φ ( U ) , [ , ] ⊲, r ) is a Lie algebra and the linear map ξ : ( Φ ( U ) , [ , ] ⊲, r ) −→ ( Φ ( U ) , [ , ] r ) , X + α X − r ( α ) + α is an isomorphism of Lie algebras. roof. Clearly ξ is bijective. Let us show that ξ preserves the Lie brackets. It is clear that, for any X , Y ∈ U , ξ ([ X , Y ] ⊲, r ) = [ ξ ( X ) , ξ ( Y )] r . Now, for any X ∈ U , α ∈ U ∗ , ξ ([ X , α ] ⊲, r ) = ξ ( − L tX α ) = r (L tX α ) − L tX α (10) = L t α X − [ X , r ( α )] − L tX α = [ X , − r ( α ) + α ] r = [ ξ ( X ) , ξ ( α )] r . On the other hand, for any α, β ∈ U ∗ , ξ ([ α, β ] ⊲, r ) = ξ ( ∆ (r)( α, β )) = ∆ (r)( α, β ) , [ ξ ( α ) , ξ ( β )] r = [ − r ( α ) + α, − r ( β ) + β ] r = [r ( α ) , r ( β )] + [ α, β ] + L t r ( α ) β − L t r ( β ) α + L t α r ( β ) − L t β r ( α ) (10) = [r ( α ) , r ( β )] + r (L t r ( β ) α ) − r (L t r ( α ) β ) + [r ( β ) , r ( α )] − [r ( α ) , r ( β )] = r ([ α, β ]) − [r ( α ) , r ( β )] = ∆ (r)( α, β ) . (cid:3) We can now transport the para-K¨ahler structure associated to r from ( Φ ( U ) , [ , ] r , h , i , K ) to Φ ( U ) via ξ and weget the following proposition. Proposition 3.8.
Let ( U , . ) be a left symmetric algebra and r = a + s ∈ U ⊗ U a quasi S -matrix. Then ( Φ ( U ) , [ , ] ⊲, r , h , i r , K r ) is a para-K¨ahler Lie algebra, where h X + α, Y + β i r = h X + α, Y + β i − s ( α, β ) and K r ( X + α ) = K ( X + α ) − ( α ) . Remark 2. ( a ) Actually, by using a similar method, we can generalize the result of Diatta [13]. Let ( g , [ , ]) be aLie algebra and r ∈ g ∧ g . Define on g ∗ and Φ ( g ) , respectively, two brackets [ , ] ∗ and [ , ] r by [ α, β ] ∗ = ad ∗ r ( α ) β − ad ∗ r ( β ) α and [ X + α, Y + β ] r = [ X , Y ] + [ α, β ] ∗ − ad tX β − ad t α Y + ad tY α + ad t β X , and [r , r] ∈ g ⊗ g ⊗ g ≃ End( g ∗ ⊗ g ∗ , g ) by [r , r]( α, β ) = r ([ α, β ] ∗ ) − [r ( α ) , r ( β )] . It is well-known that [ , ] ∗ is a Lie bracket i ff [r , r] is ad -invariant. In this case, [ , ] r is a Lie bracket. Define anew bracket on Φ ( g ) by putting [ X + α, Y + β ] ⋄ , r = [ X , Y ] + ad ∗ X β − ad ∗ Y α + [r , r]( α, β ) . By using the same argument in the proof of Proposition 3.7, one can see that ( Φ ( g ) , [ , ] ⋄ , r ) is a Lie algebra andthe linear map ξ : ( Φ ( g ) , [ , ] ⋄ , r ) −→ ( Φ ( g ) , [ , ] r ) , X + α X − r ( α ) + α is an isomorphism of Lie algebras.When [r , r] = , we recover the result of Diatta. ( b ) Let ( U , . ) be a left symmetric algebra and r = a + s ∈ U ⊗ U a quasi S -matrix. The Lie algebra ( Φ ( U ) , [ , ] ⊲, r ) is a Z -graded Lie algebra and hence L : U ∗ × U ∗ × U ∗ −→ U ∗ given byL ( α, β, γ ) = L ∗ ∆ (r)( α,β ) γ is a Lie triple system (see for instance [15, 18, 20] for the definition and the properties of Lie triple systems). . Some classes of para-K¨ahler Lie algebras In this section, we develop some methods to build para-K¨ahler Lie algebras based on the following propositionwhere we adopt the notations of the last section, in particular, Proposition 3.8.
Proposition 4.1.
Let ( U , . ) be a left symmetric algebra and r = a + s ∈ U ⊗ U which is L U -invariant. Then L( a ) = , [[r , r]] = ∆ (r) = and ( Φ ( U ) , [ , , ] ⊲ , h , i r , K r ) a para-K¨ahler Lie algebra. Moreover, the Levi-Civita product of ( Φ ( U ) , [ , , ] ⊲ , h , i r ) is ⊲ given by (8) . Proof.
Since r is L U -invariant then so a and hence L( a ) =
0. The vanishing of ∆ (r) is immediate. So we can applyProposition 3.8. To conclude, one can check easily that ⊲ is actually the Levi-Civita product of ( Φ ( U ) , [ , , ] ⊲ , h , i r ). (cid:3) As consequence of this proposition we get the following large class of para-K¨ahler Lie algebras.
Proposition 4.2.
Let ( g , h , i ) be a pseudo-Riemannian flat Lie algebra, ie., the Levi-Civita product ” . ” is left sym-metric. Denote by ♭ : g −→ g ∗ the isomorphism associated to h , i . Then ( Φ ( g ) , [ , , ] ⊲ , h , i ♭ , K ♭ ) is a para-K¨ahler Liealgebra, where h X + α, Y + β i ♭ = h X + α, Y + β i − h ♭ − ( α ) , ♭ − ( β ) i and K ♭ ( X + α ) = K ( X + α ) − ♭ − ( α ) . Moreover, the Levi-Civita product of ( Φ ( g ) , [ , , ] ⊲ , h , i ♭ ) is ⊲ given by (8) . Proof.
The Levi-Civita product defines a left symmetric algebra structure on U and r ∈ U ⊗ U defined byr( α, β ) = h ♭ − ( α ) , ♭ − ( β ) is L U -invariant and one can conclude by using Proposition 4.1. (cid:3) Let us give some methods to build pseudo-Riemannian flat Lie algebras.Let ( U , [ ., . ] , ω ) be a symplectic Lie algebra and B a nondegenerate bi-invariant bilinear symmetric form on U . Theisomorphism D defined by ω ( X , Y ) = B( D ( X ) , Y )is an invertible derivation and hence U is nilpotent (see [16]). The nondegenerate symmetric bilinear form h , i givenby h X , Y i = B( D ( X ) , D ( Y ))satisfies h X . Y , Z i + h Y , X . Z i = , where the dot designs the left symmetric product associated to ω given by (4). Thus ( U , h , i ) is a flat pseudo-Riemannian Lie algebra (see [7]). So any symplectic quadratic Lie algebra ( U , B , ω ) gives rise to a flat pseudo-Riemannian Lie algebra ( U , h , i ).More generally, let ( g , [ , ] , B ) be a quadratic Lie algebra and r ∈ g ∧ g a solution of the classical Yang-Baxterequation. The product on g ∗ given by α.β = ad ∗ r ( α ) β is left symmetric and hence induces a Lie bracket [ , ] r on g ∗ . Infact, this product is the Levi-Civita product of B ∗ (the induced bilinear nondegenerate symmetric form on g ∗ ). Thus( g ∗ , [ , ] r , B ∗ ) is a flat pseudo-Riemannian Lie algebra (see [9]).Let us give now a method to built symplectic quadratic Lie algebras.Let n ∈ N ∗ and A a vector space with a basis { e , . . . , e n } . We consider on A the product defined by e i e j = e j e i = e i + j if i + j ≦ n , e i e j = e j e i = i + j > n . The vector space A endowed with this product is a commutative and associative algebra.Let ( L , [ ., ]) be an arbitrary Lie algebra. Then the following product[ X ⊗ a , Y ⊗ b ] T : = [ X , Y ] ⊗ ab , defines a structure of Lie algebra on the vector space T : = L ⊗ A . Moreover the endomorphism δ of T defined by δ ( X ⊗ e i ) : = iX ⊗ e i X ∈ L and any i ∈ { , . . . , n } , is an invertible derivation of T .Now, on the vector space U : = T ⊕ T ∗ we will define a structure of symplectic quadratic algebra in the followingway. For any s , t ∈ T and any f , h ∈ T ∗ , put[ t + f , s + h ] U = [ t , s ] T − h ◦ ad T ( t ) + f ◦ ad T ( s ) , B( t + f , s + h ) = f ( s ) + h ( t ) , D ( t + f ) = δ ( t ) − f ◦ δ,ω ( t + f , s + h ) = B( D ( t + f ) , s + h ) . One can check easily that ( U , B , ω ) is a symplectic quadratic Lie algebra. Proposition 4.3.
Let ( g , [ , ]) be a Lie algebra, b ∈ ∧ g is a solution of the classical Yang-Baxter equation on ( g , [ , ]) , i.e. [b , b] = , and r = s + a ∈ g ∗ ⊗ g ∗ such that ad ∗ b ( α ) r = , for any α ∈ g ∗ . Then ( Φ ( g ) , [ , ] b , h , i r , K r ) is apara-K¨ahler Lie algebra, where [ X + α, Y + β ] b = ad ∗ b ( α ) β − ad ∗ b ( β ) α + [b ( α ) , Y ] + [ X , b ( β )] , h X + α, Y + β i r = h X + α, Y + β i − s ( X , Y ) and K r ( X + α ) = − K ( X + α ) − ( X ) . Moreover, the Levi-Civita product associated to ( Φ ( g ) , [ , ] b , h , i r ) is left symmetric and it is given by ( X + α ) ⊲ b ( Y + β ) = ad ∗ b ( α ) β + [b ( α ) , Y ] . Proof.
It is well-known that the product on g ∗ given by α.β = ad ∗ b ( α ) β is left symmetric and the conditionad ∗ b ( α ) r = g ∗ . So ( g ∗ , ., r) satisfies the hypothesis ofProposition 4.1 and the proposition follows. (cid:3) There is an interesting case of this situation.
Corollary 4.1.
Let ( g , [ , ]) be a Lie algebra, b ∈ ∧ g is a solution of the classical Yang-Baxter equation on ( g , [ , ]) and k ∈ g ∗ ⊗ g ∗ is the Killing form. Then ( Φ ( g ) , [ , ] b , h , i k , K k ) is a para-K¨ahler Lie algebra.
5. Hyper-para-K¨ahler Lie algebras
Hyper-para-K¨ahler Lie algebras known also as hyper-symplectic Lie algebras constitute a subclass of the class ofpara-K¨ahler Lie algebras. We will use our study in the last sections to give a new characterization of these Lie alge-bras. This characterization leads to a notion of compatibility between two left symmetric algebra structures on a givenvector space. Since a hyper-para-K¨ahler Lie algebra has a complex product structure we get also a characterization ofsuch structures.A hyper-para-K¨ahler Lie algebra is a para-K¨ahler Lie algebra ( g , h , i , K ) endowed with an endomorphism J suchthat, J = − Id g , JK = − K J , J is skew-symmetric with respect to h , i and J is invariant with respect to the Levi-Civitaproduct. According to Theorem 3.1, a para-K¨ahler Lie algebra can be identified to the phase space of Lie-extendibleleft symmetric algebras so it is natural to see how hyper-para-K¨ahler Lie algebras can be described in this sitting. Proposition 5.1.
Let ( U , . ) and ( U ∗ , . ) be a couple of Lie-extendible left symmetric algebras, ( Φ ( U ) , h , i , K ) theassociated para-K¨ahler Lie algebra and J : Φ ( U ) −→ Φ ( U ) an endomorphism. Then ( Φ ( U ) , h , i , K , J ) is ahyper-para-K¨ahler Lie algebra if and only if there exists a bilinear nondegenerate ω ∈ ∧ U ∗ such that: ( i ) for any X ∈ U , α ∈ U ∗ , JX = ♭ ( X ) and J α = − ♭ − ( α ) where ♭ : U −→ U ∗ is the isomorphism given by ♭ ( X ) = ω ( X , . ) , ( ii ) ( U , ., ω ) and ( U , ◦ , ω ) are symplectic left symmetric algebras where ◦ is given byX ◦ Y = ♭ − ( ♭ ( X ) .♭ ( Y )) . roof. From the relation JK = − K J , we deduce that for any X ∈ U , JX ∈ U ∗ and hence J defines anisomorphism ♭ : U −→ U ∗ . Moreover, from J = − Id g we deduce that J α = − ♭ − α for any α ∈ U ∗ . The skew-symmetry of J implies that ω ∈ ∧ U ∗ given by ω ( X , Y ) = ≺ ♭ ( X ) , Y ≻ is skew-symmetric and, actually is nondegenerate. Now J is invariant if and only if, for any X , Y , ∈ U and any α, β ∈ U ∗ , − L tX ( JY ) = J ( X . Y ) , X . J ( α ) = − J (L tX α ) , α. J ( X ) = − J (L t α X ) and − L t α ( J β ) = J ( α.β ) . This is equivalent toL tX ◦ ♭ + ♭ ◦ L X = ♭ − ◦ L tX + L X ◦ ♭ − = t α ◦ ♭ − + ♭ − ◦ L α = ♭ ◦ L t α + L α ◦ ♭ = , for any X ∈ U , α ∈ U ∗ . Now it is obvious that these relations are equivalent toL tX ◦ ♭ + ♭ ◦ L X = ♭ ◦ L t α + L α ◦ ♭ = , (14)for any X ∈ U , α ∈ U ∗ . One can see easily that this is equivalent to ω ( X . Y , Z ) + ω ( Y , X . Z ) = ω ( X ◦ Y , Z ) + ω ( Y , X ◦ Z ) = , for any X , Y , Z ∈ U . Thus ( U , ., ω ) and ( U , ◦ , ω ) are symplectic left symmetric algebras. The converse is obviouslytrue. (cid:3) Now, let U be a vector space, ω ∈ ∧ U ∗ nondegenerate and • , ◦ two products on U such that ( U , • , ω ) and ( U , ◦ , ω )are symplectic left symmetric algebras. Define J : Φ ( U ) −→ Φ ( U ) by J X = ♭ ( X ) and J α = − ♭ − ( α ) and denote by ♭ ( ◦ ) the product on U ∗ image by ♭ of ◦ . Let the dot denote the product on Φ ( U ) extending ( U , • ) and ( U ∗ , ♭ ( ◦ )) by (5).It is easy to see, by using (14), that for any X , Y ∈ U , α, β ∈ U ∗ ,( X + α ) . ( Y + β ) = X • Y + ♭ − ( α ) ◦ Y − (cid:16) L ◦ ♭ − ( α ) (cid:17) t β − (cid:0) L • X (cid:1) t β. (15)We have shown in Proposition 3.3 that this product is Lie-admissible i ff (7) hold. Thank to ω we can identify Φ ( U ) to T ( U ). Indeed, define ξ : T ( U ) −→ Φ ( U ) by ξ ( X , = X and ξ (0 , X ) = ♭ ( X ) . We have Ω = ξ ∗ Ω , h , i = ξ ∗ h , i , K = ξ − ◦ K ◦ ξ and J = ξ − ◦ J ◦ ξ . It is easy to check that( X , Y ) . ( Z , T ) : = ξ − ( ξ ( X , Y ) .ξ ( Z , T )) = ( X • Z , X • T ) + ( Y ◦ Z , Y ◦ T ) . (16)Now on can see easily by using (14) that, for any X ∈ U and any α ∈ U ∗ , ρ ( X , α ) = − K • , ◦ ( X , ♭ − ( α )) and ρ ∗ ( α, X ) = ♭ ◦ K • , ◦ ( X , ♭ − ( α )) ◦ ♭ − , where K • , ◦ ( X , Y ) = [L • X , L ◦ Y ] − (cid:0) L ◦ X • Y − L • Y ◦ X (cid:1) . (To distinguish between • and ◦ , we denote by L • X the left multiplication by X associated to • and so on). So by usingProposition 3.3, we get the following proposition which, actually, does not involves ω . Proposition 5.2.
Let U be a vector space and • , ◦ two left symmetric products on U. The following assertions areequivalent: The product given by (16) is Lie-admissible. For any X , Y , Z ∈ U, K • , ◦ ( X , Y ) Z = K • , ◦ ( Z , Y ) X and K • , ◦ ( X , Y ) Z = K • , ◦ ( X , Z ) Y . oreover,, the product given by (16) is left symmetric if and only if K • , ◦ vanishes identically. One can see easily that the second assertion in this proposition is equivalent to Y ◦ [ X , Z ] • − [ Y ◦ X , Z ] • − [ X , Y ◦ Z ] • = ( Z • Y ) ◦ X − ( X • Y ) ◦ Z , (17) Y • [ X , Z ] ◦ − [ Y • X , Z ] ◦ − [ X , Y • Z ] ◦ = ( Z ◦ Y ) • X − ( X ◦ Y ) • Z , (18) for any X , Y , Z ∈ U .Let us state an important formula. Let • and ◦ be two algebra structures on a vector space U . A straightforwardcomputation gives the following formula:K • + ◦ ( X , Y ) = K • ( X , Y ) + K ◦ ( X , Y ) + K • , ◦ ( X , Y ) − K • , ◦ ( Y , X ) , (19)where K x is the curvature of x . Definition 5.1.
Two left symmetric algebras structures • and ◦ on U will be called compatible if they satisfy (17) - (18) or equivalently K • , ◦ satisfies the second assertion in Proposition 5.2. The following proposition is an immediate consequence of (19).
Proposition 5.3.
Let • , ◦ be two compatible left symmetric algebra structures on U. Then for any a , b ∈ R , ( U , a • + b ◦ ) is a left symmetric algebra. Remark 3.
Let • , ◦ be two compatible left symmetric algebra structures on U. As consequence of Proposition 5.3,the bracket a [ , ] • + b [ , ] ◦ is Lie bracket and hence the two dual Poisson structures on U ∗ associated to [ , ] • and [ , ] ◦ are compatible (see for instance [21] for the definition of compatible Poisson structures). Finally, we get a characterization of hyper-para-K¨ahler Lie algebras. Actually, our study can be generalizedeasily to give a characterization of complex product structures. The characterization given in the following theoremcompletes the study of complex product structures achieved in [2].
Theorem 5.1. Let • , ◦ be two compatible left symmetric algebra structures on U. Then ( T ( U ) , K , J ) endowedwith the Lie algebra structure associated to the product given by (16) is a complex product Lie algebra. More-over, all complex product Lie algebras are obtained in this way. Let • , ◦ be two compatible left symmetric algebra structures on U and ω ∈ ∧ U ∗ such that ( U , ω, • ) and ( U , ω, ◦ ) are symplectic left symmetric algebras. Then ( T ( U ) , h , i , K , J ) endowed with the Lie algebra structureassociated to the product given by (16) is a hyper-para-K¨ahler Lie algebra. Moreover, all hyper-para-K¨ahlerLie algebras are obtained in this way. Proof.
1. To show that ( T ( U ) , K , J ) is a complex product Lie algebra it su ffi ces to show that the Nijenhuis torsion of K and J vanishes which is easy to check. Conversely, let ( g , K , J ) be a complex product Lie algebra. We have g = g ⊕ g − where g i = ker( K + i Id g ) and J defines an isomorphism φ : g −→ g − . We consider the product” . ” on g given by( u + u − ) . ( v + v − ) = u ◦ v + φ ( u ◦ φ − ( v − )) + φ − ( u − ⋆ φ ( v )) + u − ⋆ v − , where ◦ and ⋆ are the products on g and g − , respectively, given by u ◦ v = − π J [ u , Jv ] and u − ⋆ v − = − π − J [ u − , Jv − ] , where π i is the projection on g i . It was shown in [2] that ◦ , ⋆ are left symmetric and ” . ” is Lie-admissible.Put U = g , • = φ − ( ⋆ ) and define ξ : T ( U ) −→ g by ξ ( X , = X and ξ (0 , X ) = φ ( X ). We get the desiredisomorphism.2. This is a consequence of the study above. (cid:3) xample 2. Let ( U , . ) be a left symmetric algebra. Then ” . ” is compatible with itself so ( T ( U ) , K , J ) endowed withthe Lie algebra bracket associated to the left symmetric product ( X , Y ) . ( Z , T ) = ( X . Z + Y . Z , Y . T + X . T ) is a complex product Lie algebra. Moreover, if U carries ω such that ( U , ., ω ) is a symplectic left symmetric algebrathen ( T ( U ) , h , i , K , J ) is a hyper-para-K¨ahler Lie algebra. The following proposition is immediate.
Proposition 5.4.
Let • , ◦ be two compatible left symmetric algebra structures on U and ( T ( U ) , K , J ) the associatedcomplex product structure. Then the following are equivalent: ( i ) K is abelian. ( ii ) J is abelian. ( iii ) Both • and ◦ are commutative and hence associative. According to (17)-(18), two associative and commutative algebra structures • and ◦ on U are compatible if, forany X , Y , Z ∈ U , ( Z • Y ) ◦ X − ( X • Y ) ◦ Z = ( Z ◦ Y ) • X − ( X ◦ Y ) • Z = . (20)In this case, ( T ( U ) , K , J ) endowed with the bracket associated to the product given by (16) is an abelian complexproduct structure. There is similar result in [2] with a di ff erent product.
6. Quasi S -matrices on symplectic Lie algebras We have shown in Section 2 that finding the set of quasi S -matrices on a given left symmetric algebra gives a largeclass of para-K¨ahler Lie algebras. In this section, we investigate the set of quasi S -matrices with respect to the leftsymmetric product associated to a symplectic Lie algebra.Let ( g , ω ) be a symplectic Lie algebra and ♭ : g −→ g ∗ the isomorphism given by ♭ ( X ) = ω ( X , . ). The product a given by (4) is left symmetric. We associate to any endomorphism A : g −→ g the tensor YB( A ) ∈ End( g ⊗ g , g ) givenby YB( A )( X , Y ) = A [ AX , Y ] + A [ X , AY ] − [ AX , AY ] . (21)The following proposition gives an useful characterization of quasi S -matrices on ( g , a ). Proposition 6.1.
Let r ∈ g ⊗ g and a its skew-symmetric part. Put A = r ◦ ♭ and T = a ◦ ♭ . Then the followingassertions holds: ( i ) The tensor r is a quasi S -matrix of ( g , a ) if and only if T and YB( A ) are ad -invariant. ( ii ) If r is symmetric then it is a S -matrix of ( g , a ) if and only if YB( A ) = . ( iii ) If r is symmetric and invertible then it is a S -matrix of ( g , a ) if and only if A − is a derivation of the Lie algebra g . Proof.
From (4), we get that for any X ∈ g , ♭ ◦ ad X = L ∗ X ◦ ♭. (22)Moreover, we have, for any X , Y ∈ g , ∆ (r)( ♭ X , ♭ Y ) = r ([ ♭ X , ♭ Y ]) − [r ( ♭ X ) , r ( ♭ Y )] (10) = A ◦ ♭ − (cid:16) L tAY ♭ X (cid:17) − A ◦ ♭ − (cid:16) L tAX ♭ Y (cid:17) + A ◦ ♭ − (L( a )( ♭ X , ♭ Y )) − [ AX , AY ] (22) = YB( A )( X , Y ) + A ◦ ♭ − (L( a )( ♭ X , ♭ Y )) . i ) and ( ii ) holds. Now it is easy that if A is invertible then YB( A ) = A − is a derivation of the Lie algebra g and ( iii ) holds. (cid:3) Let g be a Lie algebra. The modified Yang-Baxter equation is the equationYB( A )( X , Y ) = t [ X , Y ] , for all X , Y ∈ g , (23)where t ∈ R is a fixed parameter and the unknown A is an endomorphism of g . When t = Proposition 6.2.
Let ( g , ω ) be a symplectic Lie algebra and A a solution of the modified Yang-Baxter equation whichis skew-symmetric with respect to ω . Then r = A ◦ ♭ − is a quasi S -matrix of ( g , a ) . Theorem 6.1.
Let ( g , ω ) be a symplectic Lie algebra and A : g −→ g . Put A = A s + A a where A s and A a are,respectively, the symmetric and the skew-symmetric part of A (with respect to ω ). If both YB( A ) and A s are ad -invariant, then the product ◦ on g given by X ◦ Y = X . [( A s − A a ) Y ] − ( AX ) . Y is left symmetric and ( T ( g ) , h , i A , K A ) endowed with the Lie bracket given by [( X , Y ) , ( Z , T )] A = ([ X , Z ] + YB( A )( Y , T ) , [ X , T ] + [ Z , Y ]) is a para-K¨ahler Lie algebra, where h ( X , Y ) , ( Z , T ) i A = ω ( T , X ) + ω ( Y , Z ) + ω ( A a Y , T ) and K A ( X , Y ) = ( X − AY , − Y ) , and the dot is the left symmetric product associated to ( g , ω ) . Proof.
According to Proposition 6.1, r given by r( α, β ) = − ω ( A ♭ − ( α ) , ♭ − ( β )) is a quasi S -matrix with respectto the left symmetric product associated to ω . By virtue of Corollary 3.1 and Proposition 3.8, r defines on g ∗ a leftsymmetric Lie algebra structure by (9) and ( Φ ( g ) , [ , ] ⊲, r h , i r , K r ) is a para-K¨ahler Lie algebra. We consider now thelinear isomorphism µ : T ( g ) −→ Φ ( g ), ( X , Y ) ( X , ♭ ( Y )). Thus ( T ( g ) , [ , ] µ , µ ∗ h , i r , µ − ◦ K r ◦ µ ) is a para-K¨ahlerLie algebra, [ , ] µ is a pull-back by µ of [ , ] ⊲, r . One can check easily that this bracket is the Lie bracket given in thestatement of the theorem, h , i A = µ ∗ h , i r and K A = µ − ◦ K r ◦ µ .Let us compute now the pull-back by ♭ of the left symmetric product on g ∗ given by (9). We have ≺ α, X ◦ Y ≻ = − ≺ ♭ ( X ) .♭ ( Y ) , ♭ − ( α ) ≻ (9) = − r(L t ♭ − ( α ) ♭ ( X ) , ♭ ( Y )) − r( ♭ ( X ) , ad t ♭ − ( α ) ♭ ( Y )) (22) = ω ( A [ ♭ − ( α ) , X ] , Y ) − ω ( AX , ♭ − (ad t ♭ − ( α ) ♭ ( Y ))) = ω ([ ♭ − ( α ) , X ] , A s Y ) + ω ([ X , ♭ − ( α )] , A a Y ) + ω ( Y , [ ♭ − ( α ) , AX ]) = ≺ α, X . [( A s − A a ) Y ] − ( AX ) . Y ≻ . (cid:3) Remark 4.
Actually, this theorem and Remark 2 ( b ) suggest as the following more general result. Let g be a Liealgebra and A an endomorphism of g such that YB( A ) is ad -invariant. Then, one can check that the bracket [ , ] A onT ( g ) is a Lie bracket and hence L A : g × g × g −→ g given byL A ( X , Y , Z ) = [YB( A )( X , Y ) , Z ] is a Lie triple system. Example 3.
Let ( U , . ) be a left symmetric algebra with an invertible derivation D. We know that on the vector space Φ ( U ) : = U ⊕ U ∗ we have a left symmetric structure defined by: ( X + α ) ⊲ ( Y + β ) : = X . Y − L tX β, ∀ X , Y ∈ U , α, β ∈ U ∗ . oreover ( Φ ( U ) , ⊲ , Γ ) is a symplectic left symmetric algebra. Now, it is easy to verify that the endomorphism ∆ of Φ ( U ) defined by: ∆ ( X + α ) : = D ( X ) − α ◦ D , ∀ X ∈ U , α ∈ U ∗ , is an invertible derivation of ( Φ ( U ) , ⊲ ) which is skew-symmetric with respect to Γ . We are going, in the following,to construct a left symmetric algebras with an invertible derivation. Let n ∈ N ∗ and A a vector space with a basis { e , . . . , e n } . We consider on A the product defined by:e i e j = e j e i : = e i + j if i + j ≤ n , e i e j = e j e i : = if i + j > n . The vector space A endowed with this product is a commutative associative algebra. Let ( V , ⋆ ) be a symmetric algebra,then ( U : = V ⊗ A , . ) is a left symmetric algebra where the product ”.” is defined by:v ⊗ e i . w ⊗ e j : = v ⋆ w ⊗ e i e j , ∀ v , w ∈ V , i , j ∈ { , . . . , n } . Moreover, the endomorphism D of U defined by:D ( v ⊗ e i ) : = iv ⊗ e i , ∀ v ∈ V , i ∈ { , . . . , n } , is an invertible derivation of ( U , . ) . Finally, by using the first construction, we obtain a symplectic left symmetricalgebra ( Φ ( U ) , ⊲ , Γ ) with an invertible derivation ∆ which is skew-symmetric with respect to Γ .
7. Quasi S -matrices on a left symmetric algebra U with an invariant isomorphism Θ : U −→ U ∗ In this section, we investigate the set of quasi S -matrices on a left symmetric algebra U with an invariant isomor-phism Θ : U −→ U ∗ . The most important classes are symplectic left symmetric algebras and flat pseudo-RiemannianLie algebras.Let ( U , . ) be a left symmetric algebra and Θ : U −→ U ∗ un isomorphism which is invariant, i.e., for any X ∈ U , Θ ◦ L X = L ∗ X ◦ Θ . (24)We associate to any endomorphism A : U −→ U the tensors δ ( A ) , O ( A ) ∈ End( U ⊗ U , U ) given by δ ( A )( X , Y ) = X . A ( Y ) − Y . A ( X ) − A ([ X , Y ]) and O ( A )( X , Y ) = [ AX , AY ] − ( A ( AX . Y ) − A ( AY . X )) . (25)One can see easily that O ( A ) = N A + A ◦ δ ( A ) , (26)where N A is the Nijenhuis torsion of A . The following proposition gives an useful characterization of quasi S -matricesand S -matrices on ( U , . ). The second assertion of this proposition was obtained by Bai (see Corollary 6.8 [4]). Proposition 7.1.
Let r ∈ U ⊗ U and a its skew-symmetric part. Put A = r ◦ Θ and T = a ◦ Θ . Then the followingassertions holds: ( i ) The tensor r is a quasi S -matrix of ( U , . ) if and only if T is L U -invariant and O ( A ) is L ∗ U ⊗ L ∗ U ⊗ ad -invariant. ( ii ) If r is symmetric then it is a S -matrix of ( U , . ) if and only if O ( A ) = . ( iii ) If r is symmetric and invertible then it is a S -matrix of ( U , . ) if and only if δ ( A − ) = . Proof.
We have, for any X , Y ∈ U , ∆ (r)( Θ X , Θ Y ) = r ([ Θ X , Θ Y ]) − [r ( Θ X ) , r ( Θ Y )] (10) = A ◦ Θ − (cid:16) L tAY Θ X (cid:17) − A ◦ Θ − (cid:16) L tAX Θ Y (cid:17) + A ◦ Θ − (L( a )( Θ X , Θ Y )) − [ AX , AY ] (24) = − [ AX , AY ] − A ( AY . X ) + A ( AX . Y ) + A ◦ Θ − (L( a )( Θ X , Θ Y )) = −O ( A )( X , Y ) + A ◦ Θ − (L( a )( Θ X , Θ Y )) . i ) and ( ii ) holds. Now it is easy that if A is invertible then O ( A ) = δ ( A − ) = iii ) holds. (cid:3) According to a terminology used by Bai [4], if O ( A ) = A is called un O -operator for the Lie algebra ( U , [ , ])with respect to the representation L U .There are two interesting cases:( i ) The isomorphism Θ is skew-symmetric. In this case ( U , ., ω ) is a symplectic left symmetric algebra where ω ( X , Y ) = ≺ Θ ( X ) , Y ≻ .( i ) The isomorphism Θ is symmetric. In this case ( U , ., h , i ) is a flat pseudo-Riemannian Lie algebra where h X , Y i = ≺ Θ ( X ) , Y ≻ . Proposition 7.2.
Let ( U , . ) be a left symmetric algebra and Θ : U −→ U ∗ a invariant isomorphism and r ∈ U ⊗ U.Put A = r ◦ Θ and denote by ◦ the product on U pull-back by Θ of the product on U ∗ given by (9) . Then the followingassertions holds. If Θ is skew-symmetric then, for any X , Y ∈ U,X ◦ Y = [ AX , Y ] + A ( Y . X ) + Q ( X , Y ) , (27) where Q : U × U −→ U is defined by ≺ α, Q ( X , Y ) ≻ = − ω ( δ ( A s − A a )( Θ − ( α ) , Y ) , X ) , ∀ α ∈ U ∗ , A s and A a are respectively the symmetric part and the skew-symmetric part of A with respect to the 2-form ω associated to Θ . If Θ is symmetric then, for any X , Y ∈ U,X ◦ Y = Y . AX + AX . Y − A ( Y . X ) + P ( X , Y ) , (28) where P : U × U −→ U is defined by ≺ α, P ( X , Y ) ≻ = h δ ( A s − A a )( Θ − ( α ) , Y ) , X i , ∀ α ∈ U ∗ , A s and A a are respectively the symmetric part and the skew-symmetric part of A with respect the 2-form h , i associated to Θ . If Θ is skew-symmetric and r is a quasi S -matrix then ( U , ◦ , ω ) is a symplectic left symmetric algebra if and onlyif δ ( A a ) = . Proof.
1. Suppose that Θ is skew-symmetric and define ω by ω ( X , Y ) = ≺ Θ ( X ) , Y ≻ . So, for any α, β ∈ U ∗ ,r( α, β ) = − ω ( A ◦ Θ ( α ) , β ) . We have ≺ α, X ◦ Y ≻ = ≺ α, Θ − ( Θ ( X ) . Θ ( Y )) ≻ = − ≺ Θ ( X ) . Θ ( Y ) , Θ − ( α ) ≻ (9) = − r(L t Θ − ( α ) Θ ( X ) , Θ ( Y )) − r( Θ ( X ) , ad t Θ − ( α ) Θ ( Y )) (24) = r( Θ ( Θ − ( α ) . X ) , Θ ( Y )) + ω ( AX , Θ − (cid:16) ad t Θ − ( α ) Θ ( Y ) (cid:17) ) = ω ( Y , A ( Θ − ( α ) . X )) − ω ( Y , [ Θ − ( α ) , AX ]) = ω (( A s − A a ) Y , Θ − ( α ) . X ) − ω ( Y , [ Θ − ( α ) , AX ]) = − ω ( Θ − ( α ) . ( A s − A a ) Y , X ) − ω ( Y , [ Θ − ( α ) , AX ]) = − ω ( δ (( A s − A a )( Θ − ( α ) , Y ) , X ) − ω ( Y . ( A s − A a )( Θ − ( α )) , X ) − ω (( A s − A a )([ Θ − ( α ) , Y ] , X ) − ω ( Y , [ Θ − ( α ) , AX ]) = − ω ( δ (( A s − A a )( Θ − ( α ) , Y ) , X ) + ω ( Θ − ( α ) , A ( Y . X )) − ω ([ Θ − ( α ) , Y ] , AX ) − ω ( Y , [ Θ − ( α ) , AX ]) ( a ) = − ω ( δ (( A s − A a )( Θ − ( α ) , Y ) , X ) + ≺ α, A ( Y . X )) + [ AX , Y ] ≻ . In ( a ) we have used the fact that ω is 2-cocycle with respect to the Lie bracket.18. Suppose that Θ is symmetric and define h , i by h X , Y i = ≺ Θ ( X ) , Y ≻ . So, for any α, β ∈ U ∗ ,r( α, β ) = h A ◦ Θ ( α ) , β i . We have, for any α ∈ U ∗ and X , Y ∈ U , ≺ α, X ◦ Y ≻ = ≺ α, Θ − ( Θ ( X ) . Θ ( Y )) ≻ = ≺ Θ ( X ) . Θ ( Y ) , Θ − ( α ) ≻ (9) = r(L t Θ − ( α ) Θ ( X ) , Θ ( Y )) + r( Θ ( X ) , ad t Θ − ( α ) Θ ( Y )) (24) = − r( Θ ( Θ − ( α ) . X ) , Θ ( Y )) + h AX , Θ − (cid:16) ad t Θ − ( α ) Θ ( Y ) (cid:17) i = −h Y , A ( Θ − ( α ) . X ) i + h Y , [ Θ − ( α ) , AX ] i = −h ( A s − A a ) Y , Θ − ( α ) . X i + h Y , [ Θ − ( α ) , AX ] i = h Θ − ( α ) . ( A s − A a ) Y , X i + h Y , [ Θ − ( α ) , AX ] i = h δ (( A s − A a )( Θ − ( α ) , Y ) , X i + h Y . ( A s − A a )( Θ − ( α )) , X i + h ( A s − A a )([ Θ − ( α ) , Y ] , X i + h Y , [ Θ − ( α ) , AX ] i = h δ (( A s − A a )( Θ − ( α ) , Y ) , X i − h Θ − ( α ) , A ( Y . X ) i + h [ Θ − ( α ) , Y ] , AX i + h Y , [ Θ − ( α ) , AX ] i = h δ (( A s − A a )( Θ − ( α ) , Y ) , X i − α ( A ( Y . X )) − h Y . Θ − ( α ) , AX i − h Y , AX . Θ − ( α ) i = h δ (( A s − A a )( Θ − ( α ) , Y ) , X i− ≺ α, A ( Y . X ) ≻ + ≺ α, Y . AX + AX . Y ≻ .
3. Suppose that Θ is skew-symmetric and r is a quasi S -matrix. We have, for any X , Y , Z ∈ U , ω ( X ◦ Y , Z ) + ω ( Y , X ◦ Z ) = ω ([ AX , Y ] , Z ) + ω ( A ( Y . X ) , Z ) + ω ( Q ( X , Y ) , Z ) + ω ( Y , [ AX , Z ]) + ω ( Y , A ( Z . X )) + ω ( Y , Q ( X , Z )) = − ω ( X , ( A s − A a )[ Y , Z ] − Y . ( A s − A a ) Z + Z . ( A s − A a ) Y ) − Θ ( Z )( Q ( X , Y )) + Θ ( Y )( Q ( X , Z )) = − ω ( X , δ (( A s − A a ))( Y , Z )) + ω ( δ (( A s − A a ))( Y , Z ) , X ) − ω ( δ (( A s − A a ))( Z , Y ) , X ) = − ω ( X , δ ( A s − A a ))( Y , Z )) . To conclude, remark that since r is a quasi S -matrix then its skew-symmetric part is L U -invariant and hence δ ( A s ) = (cid:3) The proof of the two following theorems is similar to the one of Theorem 6.1. The second part of Theorem 7.1 isbased on the third part of Proposition 7.2 and (26).
Theorem 7.1.
Let ( U , ., ω ) be a symplectic left symmetric algebra and A a endomorphism of U. We denote by A s andA a , respectively, the symmetric and the skew-symmetric part of A with respect to ω . The following assertions hold. If O ( A ) is L ∗ U ⊗ L ∗ U ⊗ ad -invariant and A s is L U -invariant then: ( i ) the product on U given by (27) is left symmetric, ( ii ) ( T ( U ) , [ , ] A , K A , J A ) is a complex product structure and ( T ( U ) , [ , ] A , h , i A , K A ) is a para-K¨ahler Liealgebra. If A s is L U -invariant, δ ( A a ) = and N A is L ∗ U ⊗ L ∗ U ⊗ ad -invariant then ( T ( U ) , [ , ] A , h , i A , K A , J A ) is ahyper-para-K¨ahler Lie algebra.On what above, [ , ] A , h , i A , K A , J A are given by [( X , Y ) , ( Z , T )] A = ([ X , Z ] + O ( A )( T , Y ) , X . T − Z . Y ) , J A ( X , Y ) = ( − Y + AX − A Y , X − AY ) , h ( X , Y ) , ( Z , T ) i A = ω ( T , X ) + ω ( Y , Z ) + ω ( A a Y , T ) , K A ( X , Y ) = ( X − AY , − Y ) . Theorem 7.2.
Let ( g , h , i ) be a flat pseudo-Riemannian Lie algebra and A an endomorphism of g . We denote by A s and A a , respectively, the symmetric and the skew-symmetric part of A with respect to h , i . If O ( A ) is L ∗ g ⊗ L ∗ g ⊗ ad -invariant and A a is L g -invariant then the product on g given by (28) is left symmetric. Moreover, ( T ( g ) , [ , ] A , K A , J A ) is a complex product structure and ( T ( g ) , [ , ] A , h , i A , K A ) is a para-K¨ahler Lie algebra, where [( X , Y ) , ( Z , T )] A = ([ X , Z ] + O ( A )( T , Y ) , X . T − Z . Y ) , J A ( X , Y ) = ( − Y + AX − A Y , X − AY ) , h ( X , Y ) , ( Z , T ) i A = h T , X i + h Y , Z i + h A s Y , T i , K A ( X , Y ) = ( X − AY , − Y ) . he dot here is the Levi-Civita product and L g is its associated representation. Remark 5.
As in Remark 4 we have the following more general result. Let ( U , . ) be a left symmetric algebra and Aan endomorphism of U such that O ( A ) is L ∗ U ⊗ L ∗ U ⊗ ad -invariant. Then, one can check that the bracket [ , ] A on T ( U ) is a Lie bracket and hence L A : g × g × g −→ g given byL A ( X , Y , Z ) = O ( A )( X , Y ) . Zis a Lie triple system.
8. Four dimensional hyper-para-K¨ahler Lie algebras
In this section, we determine all 4-dimensional hyper-para-K¨ahler Lie algebras, up an isomorphism. To do thatwe need first to determine two dimensional symplectic left symmetric algebras and the couples of compatible suchalgebras. Four dimensional hyper-para-K¨ahler Lie algebras were classified in [1] by a huge computation, we adopthere a new method which reduces significantly the calculations.Let ( U , ., ω ) be a symplectic left symmetric algebra. We haveR u . v = R v ◦ R u + [L u , R v ] , (29)L [ u , v ] = [L u , L v ] , (30)L u + L a u = , (31)where L a u is the adjoint of L u with respect to ω . Put U . U = span { u . v , u , v ∈ U } , D ( U . U ) = span { u . v − v . u , u , v ∈ U } , S ( U . U ) = span { u . v + v . u , u , v ∈ U } . We have clearly U . U = D ( U . U ) + S ( U . U ) and ( U . U ) ⊥ = { u ∈ U , R u = } . (32)The sign ⊥ designs the orthogonal with respect to ω . Proposition 8.1.
Let ( U , ., ω ) be an abelian symplectic left symmetric algebra. Then U . U ⊂ ( U . U ) ⊥ . Proof.
We have, for any u ∈ U , R u = L u and then we get from (29)-(30) that, for any u , v ∈ U ,R u . v = R u ◦ R v = R v ◦ R u . Moreover, R a u = − R u so R u . v = (cid:3) Proposition 8.2.
Let ( U , ., ω ) be a 2-dimensional non trivial abelian symplectic left symmetric algebra. Then thereexists a basis { e , e } of U such that ω = e ∗ ∧ e ∗ , R e = L e = and e . e = ae , a , . Proof.
We get from Proposition 8.1 that U . U = ( U . U ) ⊥ = span { e } . Choose e such that ω ( e , e ) = (cid:3) Proposition 8.3.
Let ( U , ., ω ) be a 2-dimensional non abelian symplectic left symmetric algebra. Then there exists abasis { e , e } of U such that ω = e ∗ ∧ e ∗ , e . e = , e . e = ae and e . e = − e . e = ae , a , . Proof.
We have necessary dim D ( U . U ) =
1. We distinguish two cases:20.
First case : dim U . U =
1. In this case U . U = D ( U . U ) = span { e } . If S ( U . U ) = { } then we can choose e suchthat ω ( e , e ) =
1. Since e . e , e . e ∈ S ( U . U ) then e . e = e . e =
0. Moreover, since U . U = ( U . U ) ⊥ thenR e =
0. Now e . e ∈ S ( U . U ) and then e . e =
0. Thus the product is trivial.If S ( U . U ) , { } then U . U = D ( U . U ) = S ( U . U ) = span { e } . Choose e such that ω ( e , e ) =
1. We haveR e = e . e = ae and e . e = be . The relation L [ e , e ] e = [L e , L e ]( e ) implies b = e , e ] = Second case : dim U . U =
2. In this case U . U = D ( U . U ) ⊕ S ( U . U ). Choose a basis { e , e } of U such that e ∈ D ( U . U ), e ∈ S ( U . U ) and ω ( e , e ) =
1. Since e ∈ D ( U . U ) ⊥ and e ∈ S ( U . U ) ⊥ , we get ω ( e . e + e . e , e ) = ,ω ( e . e + e . e , e ) = ω ( e . e , e ) , = − ω ( e , e . e ) = . Thus e . e = − e . e and hence [ e , e ] = e . e . So e . e = − e . e = ae . On the other hand, e . e , e . e ∈ S ( U . U ) so e . e = be and e . e = ce . Now, the relation L a e = − L e implies c = a and the relationL [ e , e ] = [L e , L e ] implies b = (cid:3) Remark 6.
From Propositions 8.2 and 8.3, one can deduce that if ( U , ., ω ) is an abelian symplectic left symmetricalgebra then D ( U . U ) = and U . U = S ( U . U ) is an ω -isotropic one dimensional vector space. However, if ( U , ., ω ) isa non abelian symplectic left symmetric algebra then U = U . U = D ( U . U ) ⊕ S ( U . U ) where D ( U . U ) and S ( U . U ) areone dimensional ω -isotropic vector spaces. This remark will play a crucial role in the proof of Theorem 8.1. Recall that two symplectic left symmetric structures ( U , ⋆, ω ) and ( U , ◦ , ω ) are called compatible if K ⋆, ◦ satisfiesthe second assertion of Proposition 5.2. It is obvious that if ( U , ⋆, ω ) is a symplectic left symmetric algebra then it iscompatible with ( U , α⋆, ω ) for any α ∈ K . We call this case trivially compatible. Theorem 8.1.
Let ( U , ⋆, ω ) and ( U , ◦ , ω ) be two symplectic left symmetric structures over a two dimensional vectorspace U. Then ( U , ⋆, ω ) and ( U , ◦ , ω ) are non trivially compatible if and only if one of the following holds: There exists a basis { e , e } such that L ⋆ e = a ! , L ⋆ e = − a − b a ! , L ◦ e = and L ◦ e = b ! ,ω = e ∗ ∧ e ∗ with a , and b , . There exists a basis { e , e } such that L ⋆ e = a ! , L ⋆ e = − a b a ! , L ◦ e = c ! , L ◦ e = − c − b c ! .ω = e ∗ ∧ e ∗ with a , , b , and c , . Proof.
The proof is based on and adequate use of the fact that the sum of two compatible symplectic left symmet-ric structures is left symmetric (see Proposition 5.3) and the use of Propositions 8.2-8.3 and Remark 6.First one can check that if ⋆ and ◦ have one of the form above then they are symplectic and compatible. Supposethat ( U , ⋆, ω ) and ( U , ◦ , ω ) are non trivially compatible. We distinguish three cases.1. Both ⋆ and ◦ are abelian. Then ⋆ + ◦ defines an abelian symplectic left symmetric algebra structure on U andhence, by virtue of Proposition 8.2, there exists a , { e , e } of U such that ω = e ∗ ∧ e ∗ e ⋆ e + e ◦ e = e ⋆ e + e ◦ e = , e ⋆ e + e ◦ e = ae . ( ∗ )Moreover, ( U ⋆ U ) ⊥ = U ⋆ U and ( U ◦ U ) ⊥ = U ◦ U .Suppose that e ◦ e ,
0. Then, from ( ∗ ) above, we get U ⋆ U = U ◦ U = span { e } , and by (32) L ⋆ e = L ◦ e = e ◦ e ,
0. So e ◦ e = e ⋆ e =
0. A same argument shows that e ◦ e = e ⋆ e = ⋆ e = L ◦ e = U ⋆ U = U ◦ U = span { e } . There existshence b , c , e ⋆ e = be and e ◦ e = ce . Finally, ◦ = bc ⋆ and this case is not possible.21. The product ⋆ is not abelian and ◦ is abelian. Then ⋆ + ◦ defines a non abelian symplectic left symmetricalgebra structure on U and hence, by virtue of Proposition 8.3, there exists a , { e , e } of U suchthat ω = e ∗ ∧ e ∗ and e ⋆ e + e ◦ e = , e ⋆ e + e ◦ e = ae , e ⋆ e + e ◦ e = − e ⋆ e − e ◦ e = ae . ( ∗∗ )Moreover, U = D ( U ⋆ U ) ⊕ S ( U ⋆ U ) and U ◦ U = S ( U ◦ U ) = ( U ◦ U ) ⊥ . By adding the two last relations in( ∗∗ ), we get e ⋆ e + e ⋆ e = e ◦ e . So e ◦ e ∈ S ( U ⋆ U ). If e ◦ e , U ◦ U and hence from the second relation in ( ∗∗ ) we deduce that e ∈ S ( U ⋆ U ) and hence U ◦ U = span { e } . So by (32) L ◦ e = e ◦ e ,
0. Thus e ◦ e = e ◦ e ,
0. We deduce from the second relation in ( ∗∗ ) that U ◦ U = span { e } and thenL ◦ e =
0. We deduce that e ⋆ e = − e ◦ e = be , e ⋆ e = − e ⋆ e = ae . From the relation L ⋆ [ e , e ] e = [L ⋆ e , L ⋆ e ] e we deduce that b = ◦ = e ◦ e = ◦ e = U ◦ U = span { e } , e ◦ e = be . We deduce that e ⋆ e = , e ⋆ e = ae − be , e ⋆ e = − e ⋆ e = ae . We get that ⋆ and ◦ satisfy the first form in the theorem.3. Both ⋆ and ◦ are non abelian. According to Proposition 5.3, ⋆ + ◦ and ⋆ −◦ define two symplectic left symmetricalgebra structures on U and one of them must be non abelian. So we can suppose that ⋆ + ◦ is non abelian byreplacing ◦ by −◦ if it is necessary. By virtue of Proposition 8.3, there exists a , { e , e } of U such that ω = e ∗ ∧ e ∗ and e ⋆ e + e ◦ e = , e ⋆ e + e ◦ e = ae , e ⋆ e + e ◦ e = − e ⋆ e − e ◦ e = ae . ( ∗ ∗ ∗ )Moreover, U = D ( U ⋆ U ) ⊕ S ( U ⋆ U ) = D ( U ◦ U ) ⊕ S ( U ◦ U ). Put v = e ⋆ e + e ⋆ e = − e ◦ e − e ◦ e . If v , S ( U ⋆ U ) and S ( U ◦ U ) so, from ( ∗ ∗ ∗ ) above, we get S ( U ⋆ U ) = S ( U ◦ U ) = span { e } .So L ⋆ e = cb ! , L ⋆ e = − c d c ! , L ◦ e = a − c − b ! , L ◦ e = c − a − d a − c ! . The relations L ⋆ [ e , e ] = [L ⋆ e , L ⋆ e ] and L ◦ [ e , e ] = [L ◦ e , L ◦ e ] ( ∗ ∗ ∗∗ )are equivalent to d = bc = b ( c − a ) which is equivalent to d = b =
0. This implies that ◦ = a − cc ⋆ . This caseis impossible and hence v = w = e ⋆ e = − e ◦ e ,
0, then it spans S ( U ⋆ U ) and S ( U ◦ U ) so, from ( ∗ ∗ ∗ ), we get S ( U ⋆ U ) = S ( U ◦ U ) = span { e } . SoL ⋆ e = cb ! , L ⋆ e = − c c ! , L ◦ e = a − c − b ! , L ◦ e = c − a a − c ! . In this case, ( ∗ ∗ ∗∗ ) imply b = ◦ = a − cc ⋆ . This case is impossible and hence w =
0. To summarize,we have shown that e ⋆ e + e ⋆ e = − e ◦ e − e ◦ e = e ⋆ e = − e ◦ e = . So L ⋆ e = c ! , L ⋆ e = − c d c ! , L ◦ e = a − c ! , L ◦ e = c − a − d a − c ! .
22e get that ⋆ and ◦ satisfy the second form in the theorem.Finally, a direct computation shows that for ⋆ and ◦ as in the first form in the theorem, we have K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = − ab ! , and for the second form K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = K ⋆, ◦ ( e , e ) = ab + bc ! . In the two cases K ⋆, ◦ satisfies the second assertion of Proposition 5.2 and the theorem is proved. (cid:3)
9. Symplectic associative algebras
In this section, we study an important subclass of the class of symplectic left symmetric algebras. In order tointroduce this subclass we begin by giving a geometric interpretation of symplectic left symmetric algebras.Let ( U , ., ω ) be a symplectic left symmetric algebra. The product being Lie-admissible, the bracket [ u , v ] = u . v − v . u is a Lie bracket on U . Moreover, since ω ( u . v , w ) + ω ( v , u . w ) = u , v , w ∈ U , a direct computation gives ω ([ u , v ] , w ) + ω ([ v , w ] , u ) + ω ([ w , u ] , v ) = , and hence ( U , [ , ] , ω ) is a symplectic Lie algebra. Let G be the simply connected Lie group associated to ( U , [ , ]).For any u ∈ U denote by u ℓ the left invariant vector field on G associated to u . The formulas ω ℓ ( u ℓ , v ℓ ) = ω ( u , v ) and ∇ u ℓ v ℓ = ( u . v ) ℓ define on G a left invariant symplectic form and a flat and torsion free left invariant connection. Moreover, ∇ issymplectic, i.e., ∇ ω ℓ =
0. The connection ∇ is right invariant i ff , for any u , v , w ∈ U ,[ u ℓ , ∇ v ℓ w ℓ ] = ∇ [ u ℓ , v ℓ ] w ℓ + ∇ v ℓ [ u ℓ , w ℓ ] . A straightforward computation shows that this relation is equivalent to the associativity of the left symmetric producton U .A symplectic associative algebra is a symplectic left symmetric algebra which is associative. We have seen that thereis a correspondence between the set of symplectic associative algebras and symplectic Lie groups endowed with abi-invariant a ffi ne structure for which the symplectic form is parallel.In what follows we will give an accurate description of symplectic associative algebras (see Theorems 9.1-9.2).Let ( U , ., ω ) be an associative symplectic algebra. Then, for any u , v ∈ U ,L uv = L u ◦ L v . Since L a u = − L u for any u ∈ U , we get L uv = L u ◦ L v = − L v ◦ L u = − L vu . (33) Proposition 9.1.
Let ( U , ., ω ) be an associative symplectic algebra. Then U = and J = U + ( U ) ⊥ is a co-isotropictwo-side ideal of U satisfying J = . roof. It is obvious that J is a co-isotropic two-side ideal. For any u , v , w ∈ U ,L uvw = L u ◦ L v ◦ L w (33) = L w ◦ L u ◦ L v = L w ◦ L uv (33) = − L uv ◦ L w = − L uvw , hence L uvw = U =
0. Recall that ( U ) ⊥ = { u ∈ U , R u = } . So U . ( U ) ⊥ = ( U ) ⊥ . ( U ) ⊥ =
0. On the other hand, for any u ∈ ( U ) ⊥ and any v , w ∈ U , u . v . w (33) = − v . u . w = , so ( U ) ⊥ . U =
0. In conclusion, J = (cid:3) According to this proposition, to study associative symplectic algebras we distinguish two cases depending on thetriviality of U or not. Model of associative symplectic algebras with U = . Let V be a vector space and ( I , s ) a symplectic vector space.Let m : V ∗ −→ V ⊙ V and n : I −→ V ⊙ V be two linear maps ( V ⊙ V is the space of bilinear symmetric forms on V ∗ ).The space U = V ⊕ I ⊕ V ∗ carries a symplectic form ω for which I and V ⊕ V ∗ are orthogonal, ω | I × I = s and for any u ∈ V , α ∈ V ∗ , ω ( α, u ) = − ω ( u , α ) = α ( u ). Define a product on U such that V ∗ . V ∗ , I . V ∗ ⊂ V by ≺ γ, α.β ≻ = m ( α )( β, γ ) and ≺ β, i .α ≻ = n ( i )( α, β ) , (34)for any α, β, γ ∈ V ∗ and i ∈ I (all the others products vanish).It is easy to see that ( U , ., ω ) is an associative symplectic algebra and U =
0. We call such algebra associativesymplectic algebra of type one.
Actually, all associative symplectic algebras with U = Theorem 9.1.
Any associative symplectic algebra with U = is isomorphic to an associative symplectic algebra oftype one. Proof.
The condition U = U ⊂ ( U ) ⊥ . Put V = U and choose I a complement of V in ( U ) ⊥ .The restriction of ω to I defines a symplectic form say s . The orthogonal I ⊥ of I is a symplectic space which contains V as a Lagrangian subspace so we can choose a Lagrangian complement W of V in I ⊥ . The linear map W −→ V ∗ , u ω ( u , . ) is an isomorphism so we can identify ( U , ω ) to V ⊕ I ⊕ V ∗ endowed with the symplectic form describedabove. Since V . U = U . V = u ∈ I , R u = U is given by (34) which achievesthe proof. (cid:3) Model of associative symplectic algebras with U , . Let V = V ⊕ V and I = I ⊕ I be two vector spaces suchthat ( I , s ), ( I , s ) are symplectic vector spaces. Put V ∗ = V ⊕ V where V and V are the annihilators of V and V respectively.Let a : V −→ V ⊙ V , b : I −→ V ⊙ V , c : I −→ V ⊙ V and d : V ∗ −→ V ⊙ V linear maps ( V ⊙ V (resp. V ⊙ V )is the space of bilinear symmetric forms on V (resp. V )). Finally, let F : V ∗ × V −→ I a bilinear map.The space U = V ⊕ I ⊕ V ∗ carries a symplectic form ω for which I and V ⊕ V ∗ are orthogonal, ω | I × I = s ⊕ s andfor any u ∈ V , α ∈ V ∗ , ω ( α, u ) = − ω ( u , α ) = α ( u ).Define now a product on U satisfying V . V , I . V , V ∗ . I ⊂ V , I . V ∗ ⊂ V , V ∗ . V ⊂ V , V ∗ . V ⊂ V ⊕ I ≺ β , v .α ≻ = a ( v )( α , β ) , v ∈ V , α , β ∈ V , ≺ β , i .α ≻ = b ( i )( α , β ) , i ∈ I , α , β ∈ V , ≺ β , α. i ≻ = s ( i , F ( α, β )) , i ∈ I , α ∈ V ∗ , β ∈ V , ≺ β, i .α ≻ = c ( i )( α, β ) , i ∈ I , α, β ∈ V ∗ ,α.β = E ( α, β ) , α ∈ V ∗ , β ∈ V ,α.β = E ( α, β ) + F ( α, β ) , α ∈ V ∗ , β ∈ V . ≺ γ, E ( α, β ) ≻ = d ( α )( β, γ ) . With this product U becomes an algebra for which the multiplication by left is symplectic. Now this product isassociative i ff , for any α ∈ V ∗ , β , γ , µ ∈ V , β ∈ V , a ( E ( α, β ))( γ , µ ) + b ( F ( α, β ))( γ , µ ) = s ( F ( β , γ ) , F ( α, µ )) , (35) a ( E ( α, β ))( γ , µ ) = s ( F ( β , γ ) , F ( α, µ )) . (36)In this case U = ff s ( F ( β , γ ) , F ( α, µ )) = s ( F ( β , γ ) , F ( α, µ )) = . When (35) and (36) hold and U , U , ., ω ) an associative symplectic algebra of type two. Theorem 9.2.
Any associative symplectic algebra with U , is isomorphic to an associative algebra of type two. Proof.
Since U = U ⊂ U then U ⊂ U ⊂ ( U ) ⊥ and U ⊂ ( U ) ⊥ ⊂ ( U ) ⊥ . Put V = U and choose a complement V of V in V = U ∩ ( U ) ⊥ . Choose I and I two subspaces of U such that U = V ⊕ I and ( U ) ⊥ = V ⊕ I . We have I ∩ I = { } , ω ( I , I ) = I , I are symplectic and I = I ⊕ I is also symplectic. Denote by s and s the restrictions of ω to I and I respectively. Now I ⊥ is symplectic and contains V as a Lagrangian subspace so wecan choose a Lagrangian subspace W of I ⊥ complement of V . The linear map W −→ V ∗ , u ω ( u , . ) realizes anisomorphism. Finally, we get the identification U = V ⊕ ( I ⊕ I ) ⊕ V ∗ with the symplectic form given by ω ( V , V ) = ω ( V , I ) = ω ( V ∗ , V ∗ ) = ω ( V ∗ , I ) = ω | I × I = ω ⊕ ω , and for any u ∈ V , α ∈ V ∗ , ω ( α, u ) = − ω ( u , α ) = α ( u ) . Denote by V and V the annihilator of V and V respectively.Let us study the product’s properties. We have shown in Proposition 9.1 that J = V ⊕ ( I ⊕ I ) satisfies J =
0. Wehave obviously that for any u ∈ V , L u = R u = u ∈ V ⊕ I , R u =
0. Since U . V ∗ ⊂ V andthe fact that the symplectic form is invariant we get U . V =
0. Since V ∗ . U ⊂ V we get from the invariance of thesymplectic form that V ∗ . V ⊂ V . So we can put ≺ β , v .α ≻ = a ( v )( α , β ) , v ∈ V , α , β ∈ V , ≺ β , i .α ≻ = b ( i )( α , β ) , i ∈ I , α , β ∈ V , ≺ β , α. i ≻ = s ( i , F ( α, β )) , i ∈ I , α ∈ V ∗ , β ∈ V , ≺ β, i .α ≻ = c ( i )( α, β ) , i ∈ I , α, β ∈ V ∗ ,α.β = E ( α, β ) , α ∈ V ∗ , β ∈ V ,α.β = E ( α, β ) + F ( α, β ) , α ∈ V ∗ , β ∈ V . a ( v ), b , c ( i ) and d ( α ) are symmetric, where d ( α )( β, γ ) = γ ( E ( α, β )).The associativity of this product is equivalent to α. ( β.γ ) = ( α.β ) .γ for any α, β, γ ∈ V ∗ . This is obviously true when γ ∈ V . So the associativity is equivalent to µ ( α. ( β .γ )) = µ (( α.β ) .γ ) and ≺ µ , α. ( β .γ ) ≻ = ≺ µ , ( α.β ) .γ ≻ for any α ∈ V ∗ , α , β , µ ∈ V and β ∈ V . Which is equivalent to s ( F ( β , γ ) , F ( α, µ )) = a ( E ( α, β ))( γ , µ ) , s ( F ( β , γ ) , F ( α, µ )) = a ( E ( α, β ))( γ , µ ) + b ( F ( α, β ))( γ , µ ) . This achieves the proof. (cid:3)
Corollary 9.1.
Let ( U , ., ω ) be an associative symplectic algebra. Then(i) If dim U = then ( U , ., ω ) is isomorphic to an associative symplectic algebra of type one of the form V ⊕ V ∗ with dim V = .(ii) If dim U = then ( U , ., ω ) is isomorphic to an associative symplectic algebra of type one either of the formV ⊕ V ∗ with dim V = or V ⊕ I ⊕ V ∗ with dim V = . A six dimensional associative algebra U , V ⊕ I ⊕ V ∗ with dim V = V = V ⊕ V .Choose a basis ( e , e ) a basis of V such e i ∈ V i and ( f , f ) a basis of I such that s ( f , f ) =
1. The equations (35)and (36) are equivalent to a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) + b ( F ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = s ( F ( e ∗ , e ∗ ) , F ( e ∗ , e ∗ )) , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = s ( F ( e ∗ , e ∗ ) , F ( e ∗ , e ∗ )) , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) + b ( F ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = s ( F ( e ∗ , e ∗ ) , F ( e ∗ , e ∗ )) , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = s ( F ( e ∗ , e ∗ ) , F ( e ∗ , e ∗ )) . Put F ( e ∗ , e ∗ ) = a f + b f and F ( e ∗ , e ∗ ) = c f + d f . So a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) + a b ( f )( e ∗ , e ∗ ) + b b ( f )( e ∗ , e ∗ ) = cb − ad , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) + c b ( f )( e ∗ , e ∗ ) + d b ( f )( e ∗ , e ∗ ) = , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = ad − cb . Which is equivalent to a b ( f )( e ∗ , e ∗ ) + b b ( f )( e ∗ , e ∗ ) = − a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) − a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) , c b ( f )( e ∗ , e ∗ ) + d b ( f )( e ∗ , e ∗ ) = − a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = , a ( E ( e ∗ , e ∗ ))( e ∗ , e ∗ ) = ad − cb . Put a ( e )( e ∗ , e ∗ ) = α , E ( e ∗ , e ∗ ) = δ = ad − cb , E ( e ∗ i , e ∗ j ) = a i j e . So b ( f )( e ∗ , e ∗ ) = − αδ − ( aa − c ( a + a )) and b ( f )( e ∗ , e ∗ ) = αδ − ( ba − d ( a + a )) . eferences [1] Andrada A., Hypersymplectic Lie algebras, J. Geometry and Physics 56 (2006), 2039-2067.[2] Andrada A. & Salamon S., Complex product structures on Lie algebras. Forum Math. 17 (2005), 261-295.[3] Bai, C., Some non-abelian phase spaces in low dimensions , Journal of Geometry and Physics, Volume 58, Issue 12, December 2008, 1752-1761.[4] Bai, C.,
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