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HYPERSYMPLECTIC STRUCTURES WITH TORSION ON LIE ALGEBROIDS P.ANTUNESANDJ.M.NUNESDACOSTA Abstract. HypersymplecticstructureswithtorsiononLiealgebroidsarein- 6 vestigated. We show that each hypersymplectic structure with torsion on a 1 Lie algebroid determines three Nijenhuis morphisms. From a contravariant 0 point of view, these structures are twisted Poisson structures. We prove the 2 existence of a one-to-one correspondence between hypersymplectic structures n with torsion and hyperka¨hler structures with torsion. We show that given a a Liealgebroidwith ahypersymplectic structure withtorsion, the deformation J oftheLiealgebroidstructurebyanyofthetransitionmorphismsdoesnotaf- 6 fectthehypersymplecticstructurewithtorsion. Wealsoshowthatifatriplet 2 of 2-forms is a hypersymplectic structure with torsion on a Lie algebroid A, then the triplet of the inverse bivectors is a hypersymplectic structure with ] torsion for a certain Lie algebroid structure on the dual A∗, and conversely. G Examplesofhypersymplecticstructureswithtorsionareincluded. D . h t a m Introduction [ Hypersymplectic structures with torsion on Lie algebroids were introduced in 2 v [4], in relation with hypersymplectic structures on Courant algebroids, when these 7 Courant algebroids are doubles of quasi-Lie and proto-Lie bialgebroids. In fact, 8 while looking for examples of hypersymplectic structures on Courant algebroids 8 we found in [4], in a natural way, hypersymplectic structures with torsion on Lie 0 algebroids. The aim of this article is to study these structures on Lie algebroids. 0 . Welookatthemasthe “symplecticversion”ofhyperkählerstructureswithtorsion 1 and also from a contravariantperspective, as twisted Poisson structures. 0 A triplet (ω ,ω ,ω ) of non-degenerate 2-forms on a Lie algebroid (A,µ) is a 5 1 2 3 1 hypersymplectic structure with torsion if the transition morphisms Ni,i = 1,2,3, : satisfyN2 = id andN dω =N dω =N dω (Definition1.1). Thesestructures v i − A 1 1 2 2 3 3 can be viewed as a generalization of hypersymplectic structures on Lie algebroids, i X a structure we have studied in [2]. We prove that they are in a one-to-one cor- r respondence with hyperkähler structures with torsion [6, 12], also known as HKT a structures, which first appeared in [7] in relation with sigma models in string the- ory. Let us briefly recall what a HKT manifold is. Let M be a hyperhermitian manifold, i.e., a manifold equipped with three complex structures N , N and N 1 2 3 satisfyingN N = N N =N andametricg compatiblewiththethreecomplex 1 2 2 1 3 − structures. If there exists a linear connection on M such that g = 0, N = 1 ∇ ∇ ∇ N = N = 0 and H, defined by H(X,Y,Z) = g(X, Z Y [Y,Z]), 2 3 Y Z ∇ ∇ ∇ −∇ − is a 3-form on M, then M is a HKT manifold. In [6] it was proved that the condition that H be a 3-form can be replaced by the following equivalent require- ment: N dω = N dω = N dω , where ω , ω and ω are the associated Kähler 1 1 2 2 3 3 1 2 3 forms. Later, in [12], the authors showed that the assumption that N , N and 1 2 N are Nijenhuis can be removed from the definition of HKT manifold, because 3 the equalities N dω = N dω = N dω imply the vanishing of the Nijenhuis tor- 1 1 2 2 3 3 sion of the morphisms N , N and N . Thus, the definition of a HKT manifold 1 2 3 1 2 P.ANTUNESANDJ.M.NUNESDACOSTA can be simplified, requiring that it is an almost hyperhermitian manifold satisfying N dω = N dω = N dω . Here, we extend the notion of HKT structure to 1 1 2 2 3 3 Lie algebroids. Our definition of hypersymplectic structure with torsion and HKT structureismoregeneralthantheusualone,sinceweconsiderthecasesof(almost) complex and para-complex morphisms N . We also look at hypersymplectic struc- i tures with torsion on Lie algebroids from a different perspective, by presenting an alternativedefinitionintermsofbivectorsinsteadof2-forms(Theorem2.2). These bivectors are twisted Poisson, also known as Poisson with a 3-form background [15]. Moreover,we provethat the almostcomplex morphisms N , N and N , that 1 2 3 are constructed out of the 2-forms and the twisted Poisson bivectors, are in fact Nijenhuis morphisms (Theorem 7.1). In other words, a hypersymplectic structure with torsionon a Lie algebroidA determines three Nijenhuis morphisms and three twisted Poisson bivectors. It is well known [8] that if (A,µ) is a Lie algebroid and N is a Nijenhuis morphism, the deformation(in a certainsense)ofµ by N yields a newLiealgebroidstructureonA,thatwedenotebyµ . Ontheotherhand,ifaLie N algebroid (A,µ) is equipped with a nondegenerate twisted Poisson bivector π, the dualvectorbundle A inherits aLiealgebroidstructuregivenby µ +1 ω,[π,π] , ∗ π 2{ } where ω is the inverse of π (Proposition 7.6). So, two natural questions arise. 1. If (ω ,ω ,ω ) is a hypersymplectic structurewith torsion on a Lie algebroid 1 2 3 (A,µ), does it remain a hypersymplectic structure with torsion on the Lie algebroid (A,µ ), for i=1,2,3? Ni InTheorem7.5weanswerthisquestionpositively,showingthat(ω ,ω ,ω ) 1 2 3 is a hypersymplectic structure with torsion on (A,µ) if and only if it is a hypersymplectic structure with torsion on (A,µ ), for i=1,2,3. Ni 2. Does a hypersymplectic structure with torsion on a Lie algebroid A induce a hypersymplectic structure with torsion on the Lie algebroid A ? ∗ The answer is given in Theorem 7.8, where we prove that (ω ,ω ,ω ) is 1 2 3 ahypersymplecticstructurewithtorsionon(A,µ)ifandonlyif(π ,π ,π ) 1 2 3 is ahypersymplecticstructurewithtorsionon A , µ 1 ω ,[π ,π ] , ∗ − πi − 2{ i i i } i=1,2,3, where πi is the inverse of ωi. (cid:0) (cid:1) Question2 was answeredin [3] for the caseof hypersymplectic structures(with- out torsion) on Lie algebroids. ε In Section 1 we define -hypersymplectic structures with torsion on a Lie algebroid and present some of their properties. In Section 2 we characterize them in terms of twisted Poisson bivectors. The structure induced on the base manifold of a Lie algebroid with a hypersymplectic structure with torsion is described in Section 3. In Section 4 we introduce the notion of hyperkähler structure with torsion on a Lie algebroid and prove the existence of a one-to-one correspondence between hypersymplectic structures with torsion and hyperkähler structures with torsion on a Lie algebroid (Theorem 4.5). Section 5 contains three examples of hypersymplectic structures with torsion on R8, on su(3) and on the tangent bundle of S3 (S1)5, respectively. In Section 6 we start by recalling the definition × of (para-)hypersymplectic structures on a pre-Courant algebroid. Then, we con- centrate on pre-Courant structures on the vector bundle A A , to study the ∗ ⊕ Nijenhuis torsion of an endomorphism of A A of type = N ( N ), with ∗ N ∗ ⊕ D ⊕ − N : A A. Namely, we establish relations between the Nijenhuis torsion of N → D and the Nijenhuis torsion of N (Propositions 6.2 and 6.4). In Proposition 6.5 we HYPERSYMPLECTIC STRUCTURES WITH TORSION ON LIE ALGEBROIDS 3 show how the Frölicher-Nijenhuis bracket on a Lie algebroid can be seen in the pre-Courant algebroid setting. Section 7 contains the main results of the paper. Givena(para-)hypersymplecticstructurewithtorsiononaLiealgebroid,weprove that the transition morphisms are Nijenhuis (Theorem 7.1) and pairwise compatible with respect to the Frölicher-Nijenhuis bracket (Proposition 7.2), and that the twisted Poisson bivectors are compatible with respect to the Schouten-Nijenhuis bracket of the Lie algebroid (Proposition 7.3). The results of [4] are extensively used in the proofs of Section 7. Mostofthecomputationsalongthepaperaredoneusingthebigbracket[9]. For furtherdetailsonLieandpre-Courantalgebroidsinthesupergeometricsetting,see [13, 14, 17, 16]. Notation: We consider 1,2 and 3 as the representative elements of the equivalence classes of Z , i.e., Z := [1],[2],[3] . Along the paper, although we omit the 3 3 brackets, and write i instea{d of [i], th}e indices are elements in Z :=Z/3Z. 3 1. Hypersymplectic structures with torsion Let (A,µ) be a Lie algebroid and take three non-degenerate 2-forms ω ,ω and 1 2 ω Γ( 2A ) with inverse π ,π and π Γ( 2A), respectively. We define the 3 ∗ 1 2 3 ∈ ∧ ∈ ∧ transition morphisms N ,N and N :A A, by setting 1 2 3 → (1) N :=π♯ ω♭ , i Z . i i−1◦ i+1 ∈ 3 In (1), we consider the usual vector bundle maps π# : A A and ω♭ : A A , ∗ ∗ → → associated to a bivector π Γ( 2A) and a 2-form ω Γ( 2A ), respectively, ∗ ∈ ∈ which are given by β,π#(α) =Vπ(α,β) and ω♭(X),Y = ω(VX,Y), for all α,β h i h i ∈ Γ(A ) and X,Y Γ(A). ∗ ∈ In what follows we shall consider the parameters ε = 1,i = 1,2,3, and the i ε ± triplet =(ε ,ε ,ε ). 1 2 3 Definition 1.1. Atriplet(ω ,ω ,ω )ofnon-degenerate2-formsonaLiealgebroid 1 2 3 ε (A,µ) is an -hypersymplectic structure with torsion if (2) N 2 =ε id , i=1,2,3, and ε N dω =ε N dω =ε N dω , i i A 2 1 1 3 2 2 1 3 3 where N , i=1,2,3, is given by (1) and i N dω (X,Y,Z):=dω (N X,N Y,N Z), for all X,Y,Z Γ(A). i i i i i i ∈ When the non-degenerate 2-forms ω ,ω and ω are closed, so that they are 1 2 3 symplectic forms and ε N dω = ε N dω = ε N dω is trivially satisfied, the 2 1 1 3 2 2 1 3 3 ε triplet (ω ,ω ,ω ) is an -hypersymplectic structure on (A,µ) [2]. 1 2 3 ε Having an -hypersymplectic structure with torsion (ω ,ω ,ω ) on a Lie alge- 1 2 3 broid A over M, we define a map (3) g :A A R, g(X,Y)= g♭(X),Y , × −→ h i where g♭ :A A is the vector bundle morphism defined by ∗ −→ (4) g♭ :=ε ε ω ♭ π ♯ ω ♭. 3 2 3 1 2 ◦ ◦ The definition of g♭ is not affected by a circular permutation of the indices in (4), that is, (5) g♭ =ε ε ω ♭ π ♯ ω ♭, i Z . i 1 i+1 i 1 i i+1 3 − − ◦ ◦ ∈ Moreover, (6) (g♭)∗ = ε1ε2ε3 g♭, − 4 P.ANTUNESANDJ.M.NUNESDACOSTA which means that g is symmetric or skew-symmetric, depending on the sign of ε ε ε . An important property of g is the following one: 1 2 3 (7) g(N X,N Y)=ε ε g(X,Y), X,Y Γ(A). i i i 1 i+1 − ∈ Noticethatg♭isinvertibleand,usingitsinverse,wemaydefineamapg 1 :A A R, − ∗ ∗ × −→ by setting (8) g 1(α,β):= β,(g♭) 1(α) , − − h i for all α,β Γ(A ). ∗ ∈ ε All the algebraic properties of -hypersymplectic structures on Lie algebroids, ε provedinPropositions3.7,3.9and3.10in[2],holdinthecaseof -hypersymplectic structures with torsion. 2. The contravariant approach In this section we give a contravariantcharacterization of an ε-hypersymplectic structure with torsion on a Lie algebroid. We shall need the next lemma, that can be easily proved. Lemma 2.1. Let ω be a non-degenerate 2-form on a Lie algebroid (A,µ), with inverse π. Then, 1 (9) [π,π]=dω π♯(),π♯(),π♯() 2 · · · (cid:0) (cid:1) or, equivalently, 1 (10) dω = [π,π] ω♭(),ω♭(),ω♭() , 2 (cid:16) · · · (cid:17) where [, ] stands for the Schouten-Nijenhuis bracket of multivectors on (A,µ). · · Equation (9) means that π is a twisted Poisson bivector on (A,µ), also known as Poissonbivector with the 3-form background dω [15]. Let(ω ,ω ,ω )be anε-hypersymplecticstructurewithtorsionon(A,µ)andlet 1 2 3 us denote by H the 3-form H :=ε N dω =ε N dω =ε N dω . 2 1 1 3 2 2 1 3 3 So, for all X,Y,Z Γ(A), ∈ (11) H(X,Y,Z)=ε dω (N X,N Y,N Z), i=1,2,3, i+1 i i i i or, equivalently, (12) dω (X,Y,Z)=ε ε H(N X,N Y,N Z), i=1,2,3. i i i+1 i i i Now we prove the main result of this section, that gives a characterization of ε-hypersymplectic structures with torsion on a Lie algebroid. Theorem 2.2. A triplet (ω ,ω ,ω ) of non-degenerate 2-forms on a Lie algebroid 1 2 3 (A,µ) with inverses π ,π and π , respectively, is an ε-hypersymplectic structure 1 2 3 with torsion on A if and only if N 2 =ε id , i=1,2,3, and ε [π ,π ]=ε [π ,π ]=ε [π ,π ], i i A 1 1 1 2 2 2 3 3 3 with N given by (1). i Proof. Assume that ε N dω =ε N dω =ε N dω . Using (9), (12) and formula 2 1 1 3 2 2 1 3 3 N π♯ =ε ε (g 1)♯ (see [2]), we have i◦ i i−1 i − ε [π ,π ] = 2ε dω π♯(),π♯(),π♯() i i i i i(cid:16) i · i · i · (cid:17) = 2ε1ε2ε3H (g−1)♯(),(g−1)♯(),(g−1)♯() , · · · (cid:0) (cid:1) HYPERSYMPLECTIC STRUCTURES WITH TORSION ON LIE ALGEBROIDS 5 for i Z ; thus, 3 ∈ ε [π ,π ]=ε [π ,π ]=ε [π ,π ]. 1 1 1 2 2 2 3 3 3 Conversely, assume that ε [π ,π ] = ε [π ,π ] = ε [π ,π ] and let us set ψ := 1 1 1 2 2 2 3 3 3 1ε [π ,π ]. Then, using (10) and formula ω♭ N =ε g♭ (see [2]), we get 2 i i i i ◦ i i−1 1 ε dω (N (),N (),N ()) = ε [π ,π ] ω♭ N (),ω♭ N (),ω♭ N () i+1 i i · i · i · 2 i+1 i i (cid:16) i ◦ i · i ◦ i · i ◦ i · (cid:17) = ε ε ε ψ g♭(),g♭(),g♭() . 1 2 3 (cid:16) · · · (cid:17) Therefore, we conclude ε N dω =ε N dω =ε N dω . 2 1 1 3 2 2 1 3 3 (cid:3) Remark 2.3. When there is an ε-hypersymplectic structure with torsion on a Lie algebroid(A,µ), this Lie algebroidis equipped with a triplet (π ,π ,π ) of twisted 1 2 3 Poisson bivectors that share, eventually up to a sign, the same obstruction to be Poisson. This obstruction is denoted by ψ in the proof of Theorem 2.2. 3. Structures induced on the base manifold Itiswellknownthatasymplecticstructureω onaLiealgebroidA M induces → a Poisson structure on the base manifold M. The Poisson bivector π on M is M defined by (13) π♯ =ρ π♯ ρ∗, M ◦ ◦ where ρ is the anchor map and π Γ( 2A) is the Poisson bivector on A which is ∈ ∧ the inverse of ω. In the case where ω is non-degenerate but not necessarily closed, so that π defines a Poisson structure with the 3-form background dω on the Lie algebroid A (see (9)), the base manifold M has an induced Poisson structure with a 3-formbackground,providedthatdω Γ( 3A ) is the pull-backby ρ ofa closed ∗ ∈ ∧ 3-form φ Ω3(M). In fact, if dω =ρ (φ ) then, a straightforwardcomputation M ∈ ∗ M gives [π,π]=2dω π♯(),π♯(),π♯() · · · which implies that (cid:0) (cid:1) [π ,π ] =2φ π♯ (),π♯ (),π♯ () , M M M M M · M · M · (cid:0) (cid:1) where π is given by (13) and [, ] stands for the Schouten-Nijenhuis bracket of M · ·M multivectors on M. The next proposition is now immediate. Proposition 3.1. Let (ω ,ω ,ω ) be an ε-hypersymplectic structure with torsion 1 2 3 on a Lie algebroid (A,µ) over M with inverses π ,π and π , respectively. Let 1 2 3 ρ be the anchor map of A. If dω = ρ (φi ), i = 1,2,3, with φi a closed 3- i ∗ M M form on the manifold M, then M is equipped with three twisted Poisson bivectors π1 ,π2 and π3 , defined by (13); the 3-forms background are φ1 , φ2 and φ3 , M M M M M M respectively. Moreover, the three bivectors on M share, eventually up to a sign, the same obstruction to be Poisson. We should stress that if ω = ρ (ωi ) for some 2-forms ωi on M, i = 1,2,3, i ∗ M M then dω = ρ (δωi ) with δ the De Rham differential on M. So, the assumptions i ∗ M of Proposition 3.1 are satisfied and [πi ,πi ] =2δωi (πi )♯(),(πi )♯(),(πi )♯() , M M M M M · M · M · i=1,2,3. However,althoughthebiv(cid:0)ectorsπ1 ,π2 andπ3 onM(cid:1)satisfyε [π1 ,π1 ] = M M M 1 M M M ε [π2 ,π2 ] = ε [π3 ,π3 ] , in general, the triplet (ω1 ,ω2 ,ω3 ) does not define 2 M M M 3 M M M M M M 6 P.ANTUNESANDJ.M.NUNESDACOSTA an ε-hypersymplectic structure with torsion on M, because ωi is not the inverse M of πi . M 4. Hypersymplectic structures with torsion versus hyperka¨hler structures with torsion ε In this section we consider -hypersymplectic structures with torsion such that ε ε ε = 1 and we prove that these structures are in one-to-one correspondence 1 2 3 − with (para-)hyperkähler structures with torsion, a notion we shall define later. ε First, let us consider two different cases of an -hypersymplectic structure with torsion such that ε ε ε = 1. 1 2 3 − Definition4.1. [4]Let(ω ,ω ,ω )beanε-hypersymplecticstructurewithtorsion 1 2 3 on a Lie algebroid (A,µ), such that ε ε ε = 1. 1 2 3 − If ε = ε = ε = 1, then (ω ,ω ,ω ) is said to be a hypersymplectic 1 2 3 1 2 3 • − structure with torsion on A. Otherwise, (ω ,ω ,ω ) is said to be a para-hypersymplectic structure with 1 2 3 • torsion on A. For a (para-)hypersymplectic structure with torsion on a Lie algebroid (A,µ), the morphism g♭ defined by (4) determines a pseudo-metric on (A,µ) [2]. Hyperkähler structures with torsion on manifolds were introduced in [7] and studied in [6] and in [12]. The definition extends to the Lie algebroid setting in a natural way. Let (A,µ) be a Lie algebroid and consider a map g:A A R and endomorphisms I ,I ,I :A A such that, for all i Z and X,Y ×Γ(A→), 1 2 3 3 → ∈ ∈ i) g is a pseudo-metric; ii) I2 =ε id , where ε = 1 and ε ε ε = 1; i i A i ± 1 2 3 − (14) iii) I =ε ε I I ; 3 1 2 1 2 ◦ iv) g(I X,I Y)=ε ε g(X,Y); i i i 1 i+1 − v) ε I dω =ε I dω =ε I dω , 2 1 1 3 2 2 1 3 3 where the 2-forms ω ,i=1,2,3, which are called the Kähler forms, are defined by i (15) ω♭ =ε ε g♭ I . i i i−1 ◦ i Definition 4.2. Aquadruple(g,I ,I ,I )satisfying(14)i) v),onaLiealgebroid 1 2 3 − (A,µ), is a hyperka¨hler structure with torsion on A if ε =ε =ε = 1; 1 2 3 • − para-hyperka¨hler structure with torsion on A, otherwise. • Remark 4.3. Most authors consider that a (para-)hyperkähler structure, with or withouttorsion,isequippedwithapositivedefinitemetric,whileforusg:A A R is a pseudo-metric, i.e., it is symmetric and non-degenerate. × → Note that, because ε ε ε = 1, on a (para-)hyperkählerstructure with torsion 1 2 3 − (g,I ,I ,I ), we always have I I = I I , for all i,j 1,2,3 ,i = j (see 1 2 3 i j j i ◦ − ◦ ∈ { } 6 Proposition 3.9 in [2]). Thenextlemmaestablishesarelationbetweenthepseudo-metricandtheKähler formsofa (para-)hyperkählerstructurewith torsionona Lie algebroid. Namely,it is shown that the pseudo-metric satisfies an equation similar to (5). Lemma 4.4. Let (g,I ,I ,I ) be a (para-)hyperka¨hler structure with torsion on a 1 2 3 Lie algebroid with associated Kähler forms ω , ω and ω . Then, 1 2 3 g♭ =ε ε ω ♭ π ♯ ω ♭, i Z , i 1 i+1 i 1 i i+1 3 − − ◦ ◦ ∈ where π is the inverse of ω . i i HYPERSYMPLECTIC STRUCTURES WITH TORSION ON LIE ALGEBROIDS 7 Proof. It is enough to prove that g♭ = ε ε ω ♭ π ♯ ω ♭. From (15) we have, on 1 3 1 2 3 ◦ ◦ one hand, (16) g♭ =ε ω♭ I =ε ω♭ I I 2 3◦ 3 1 3◦ 1◦ 2 and, on the other hand, (17) g♭ =ε ω♭ I . 1 2◦ 2 From (16) and (17), we get ω♭ I =ω♭ I =π♯ ω♭ I =ε π♯ ω♭, 3◦ 1 2 ⇔ 1 3◦ 2 ⇔ 1 1 2◦ 3 where we used (I ) 1 =ε I , and so, g♭ =ε ω♭ I yields 1 − 1 1 3 1◦ 1 g♭ =ε ε ω ♭ π ♯ ω ♭. 1 3 1 2 3 ◦ ◦ (cid:3) At this point, we shall see that (para-)hypersymplectic structures with torsion and(para-)hyperkählerstructureswithtorsiononaLiealgebroidareinone-to-one correspondence. Theorem 4.5. Let (A,µ) be a Lie algebroid. There exists a one-to-one correspon- dencebetween(para-)hypersymplecticstructureswithtorsionand(para-)hyperka¨hler structures with torsion on (A,µ). Proof. Let(ω ,ω ,ω )bea(para-)hypersymplecticstructurewithtorsiononAand 1 2 3 consider the endomorphisms N ,N ,N and g♭ given by (1) and (4), respectively. 1 2 3 SincetheequalitiesN N =ε ε N = N N holdandgsatisfiesω♭ =ε ε g♭ 1◦ 2 1 2 3 − 2◦ 1 i i i−1 ◦ N (see [2]), the quadruple (g,N ,N ,N ) is a (para-)hyperkähler structure with i 1 2 3 torsion on (A,µ) and its Kähler forms are ω ,ω and ω . 1 2 3 Conversely, let us take a (para-)hyperkähler structure with torsion (g,I ,I ,I ) 1 2 3 on A and consider the associated Kähler forms ω , ω and ω , given by (15). We 1 2 3 claimthat (ω ,ω ,ω ) is a (para-)hypersymplecticstructure with torsionon A. To 1 2 3 prove this, it is enough to show that I ,I and I are the transition morphisms 1 2 3 of (ω ,ω ,ω ), defined by (1). From ω♭ = ε ε g♭ I (definition of the Kähler forms1), w2e g3et g♭ =ε ω♭ I or, usingiLemmi ai−41.4,◦ i i−1 i ◦ i ε ω♭ π♯ ω♭ =ω♭ I , i i ◦ i+1◦ i−1 i ◦ i which is equivalent to I =ε π♯ ω♭ =π♯ ω♭ , i i i+1◦ i−1 i−1◦ i+1 where we used (I ) 1 =ε I in the last equality. (cid:3) i − i i As a consequence of Theorem4.5, if we pick a (para-)hypersymplectic structure with torsion (ω ,ω ,ω ) on (A,µ) and consider the (para-)hyperkähler structure 1 2 3 withtorsion(g,N ,N ,N )givenbyTheorem4.5then,the(para-)hypersymplectic 1 2 3 structure with torsion that corresponds to (g,N ,N ,N ) via Theorem 4.5, is the 1 2 3 initial one, i.e., (ω ,ω ,ω ). 1 2 3 5. Examples We present, in this section, three examples of hypersymplectic structures with torsion. The first two examples are on Lie algebras, viewed as Lie algebroids over a point, and the third is on the tangent Lie algebroid to a manifold. The first example is inspired from [6]. Let A ,A ,B ,B ,Z,C ,C ,C be a 1 2 1 2 1 2 3 basis for R8 and let us consider the Lie algebra{structure defined by } [A ,B ]=Z, [A ,B ]= Z 1 1 2 2 − 8 P.ANTUNESANDJ.M.NUNESDACOSTA and the remaining brackets vanish. We denote by a ,a ,b ,b ,z,c ,c ,c the 1 2 1 2 1 2 3 dual basis and we define a triplet (ω ,ω ,ω ) of 2-for{ms on R8 by setting } 1 2 3 ω =a b +a b +zc +c c ; 1 1 1 2 2 1 2 3 ω = a a +b b zc +c c ; 2 1 2 1 2 2 1 3 − − ω = a b +a b zc c c . 3 1 2 2 1 3 1 2 − − − Theirmatrixrepresentationonthebasis A ,A ,B ,B ,Z,C ,C ,C anditsdual 1 2 1 2 1 2 3 { } is the following: 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 −  0 0 0 1 0 0 0 0   1 0 0 0 0 0 0 0  1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0  −     0 1 0 0 0 0 0 0   0 0 1 0 0 0 0 0  Mω1 = 0 −0 0 0 0 1 0 0 ,Mω2 = 0 0 −0 0 0 0 1 0     −   0 0 0 0 1 0 0 0   0 0 0 0 0 0 0 1   −     0 0 0 0 0 0 0 1   0 0 0 0 1 0 0 0       0 0 0 0 0 0 1 0   0 0 0 0 0 1 0 0   −   −  and 0 0 0 1 0 0 0 0 −  0 0 1 0 0 0 0 0  0 1 0 0 0 0 0 0  −   1 0 0 0 0 0 0 0  Mω3 = 0 0 0 0 0 0 0 1 .  −   0 0 0 0 0 0 1 0   −   0 0 0 0 0 1 0 0     0 0 0 0 1 0 0 0    The 2-forms ω , ω and ω are non-degenerate and since 1 2 3 (18) dω = a b c +a b c , dω =a b c a b c , dω =a b c a b c , 1 1 1 1 2 2 1 2 1 1 2 2 2 2 3 1 1 3 2 2 3 − − − they are not closed. The transition morphisms N , N and N , given by (1), 1 2 3 correspond to the following matrices in the considered basis: = , = , = . MN1 −Mω1 MN2 −Mω2 MN3 −Mω3 Using (18), we have dω (N (),N (),N ())=dω (N (),N (),N ())=dω (N (),N (),N ())=a b z a b z, 1 1 1 1 2 2 2 2 3 3 3 3 1 1 2 2 · · · · · · · · · − so that the triplet (ω ,ω ,ω ) is a hypersymplectic structure with torsion on R8. 1 2 3 Thepseudo-metricgdeterminedby(ω1,ω2,ω3),definedby(4),issimplyg= idR8. − Next, we address an explicit example [11] of an hypersymplectic structure with torsion on the Lie algebra su(3) of the Lie group SU(3). WewriteE fortheelementary3 3-matrixwith1atposition(p,q)andconsider pq × the basis of su(3) consisting of eight complex matrices: A =i(E E ), A =i(E E ), 1 11 22 2 22 33 − − B =E E , C =i(E +E ), pq pq qp pq pq qp − HYPERSYMPLECTIC STRUCTURES WITH TORSION ON LIE ALGEBROIDS 9 wherep,q 1,2,3 suchthatp<q. We denote by a ,a ,...,c the dualbasis 1 2 23 ∈{ } { } and define a triplet of 2-forms on SU(3) by setting √3 ω = a a +b c +b c b c ; 1 1 2 12 12 13 13 23 23 − 2 − √3 1 ω = a b a c + a c b b +c c ; 2 2 12 1 12 2 12 13 23 13 23 2 − 2 − √3 1 ω = a c +a b a b +b c +b c . 3 2 12 1 12 2 12 13 23 23 13 2 − 2 The2-formshaveamatrixrepresentation,onthebasis(A ,A ,B ,B ,B ,C ,C ,C ) 1 2 12 13 23 12 13 23 and its dual, given by 0 √3 0 0 0 0 0 0 0 0 0 0 0 1 0 0  √3 −02 0 0 0 0 0 0   0 0 √3 0 0 −1 0 0  2 2 2 0 0 0 0 0 1 0 0 0 √3 0 0 0 0 0 0    − 2   0 0 0 0 0 0 1 0   0 0 0 0 1 0 0 0  Mω1 = 0 0 0 0 0 0 0 1 ,Mω2 = 0 0 0 1 −0 0 0 0   −     0 0 1 0 0 0 0 0   1 1 0 0 0 0 0 0   −   −2   0 0 0 1 0 0 0 0   0 0 0 0 0 0 0 1   −   −   0 0 0 0 1 0 0 0   0 0 0 0 0 0 1 0      and 0 0 1 0 0 0 0 0  0 0 1 0 0 √3 0 0  −2 2 1 1 0 0 0 0 0 0  − 2   0 0 0 0 0 0 0 1  Mω3 = 0 0 0 0 0 0 1 0 .    0 √3 0 0 0 0 0 0   − 2   0 0 0 0 1 0 0 0   −   0 0 0 1 0 0 0 0   −  These 2-forms are not closed; for example, we have dω = √3a (b c +b c )+√3a (b c 1 1 13 13 23 23 2 12 12 − +b c ) b b c b b c b b c c c c . 13 13 12 13 23 12 23 13 13 23 12 12 13 23 − − − − ThetransitionmorphismsN ,N andN ,givenby(1),correspondtothefollowing 1 2 3 matrices in the considered basis: 1 2 0 0 0 0 0 0 0 0 1 0 0 1 0 0 −√3 √3 −√3  2 1 0 0 0 0 0 0   0 0 2 0 0 0 0 0  −√3 √3 −√3 MN1 = 000000 000000 001000 000100 000001 −000001 −000001 100000 ,MN2 = −000001 √00002213 000000 −000001 001000 000000 000001 000100   −   −  10 P.ANTUNESANDJ.M.NUNESDACOSTA and 0 0 1 0 0 1 0 0 − −√3  0 0 0 0 0 2 0 0  −√3  1 −21 0 0 0 0 0 0  MN3 = 00 00 00 00 00 00 01 −1 .  0 √3 0 0 0 0 −0 0   2   0 0 0 0 1 0 0 0     0 0 0 1 0 0 0 0    An easy computation gives dω (N (),N (),N ())=dω (N (),N (),N ())=dω (N (),N (),N ()) 1 1 1 1 2 2 2 2 3 3 3 3 · · · · · · · · · = a b c +a b c 2a b c a b c 2a b c 1 13 13 1 23 23 1 12 12 2 13 13 2 23 23 − − − − +a b c +b c c +b c c +b c c +b b b , 2 12 12 23 12 13 13 12 23 12 13 23 12 13 23 whichshowsthatthetriplet(ω ,ω ,ω )isahypersymplecticstructurewithtorsion 1 2 3 on su(3). Finally, the pseudo-metric is given by 1 1 0 0 0 0 0 0 − 2  1 1 0 0 0 0 0 0  2 − 0 0 1 0 0 0 0 0  −   0 0 0 1 0 0 0 0  Mg = 0 0 0 −0 1 0 0 0 .  −   0 0 0 0 0 1 0 0   −   0 0 0 0 0 0 1 0   −   0 0 0 0 0 0 0 1   −  In the third example, which is taken form [12], we describe a hypersymplectic structureontheLiealgebroidtangenttothemanifoldM =S3 (S1)5. Thesphere × S3 is identified with the Lie group Sp(1). In its Lie algebra sp(1) we consider a basis A ,A ,A and the brackets 2 3 4 { } [A ,A ]=2A , [A ,A ]=2A , [A ,A ]=2A . 2 3 4 3 4 2 4 2 3 Let a ,a ,a be the dual basis and let us consider a basis a ,a ,a ,a ,a of 2 3 4 1 5 6 7 8 { } { } 1-forms on (S1)5. We define a triplet (ω ,ω ,ω ) of 2-forms on M by setting, on 1 2 3 the basis a ,a ,a ,a ,a ,a ,a ,a , 2 3 4 1 5 6 7 8 { } ω =a a +a a +a a +a a ; 1 2 1 4 3 6 5 8 7 ω =a a +a a +a a +a a ; 2 3 1 2 4 7 5 6 8 ω =a a +a a +a a +a a . 3 4 1 3 2 8 5 7 6 Their matrix representation on the considered basis is given by 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0  0 0 1 0 0 0 0 0   0 0 0 1 0 0 0 0  − 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0    −   1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0  Mω1 = −0 0 0 0 0 1 0 0 ,Mω2 = 0 −0 0 0 0 0 1 0   −   −   0 0 0 0 1 0 0 0   0 0 0 0 0 0 0 1       0 0 0 0 0 0 0 1   0 0 0 0 1 0 0 0   −     0 0 0 0 0 0 1 0   0 0 0 0 0 1 0 0     − 