Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures
aa r X i v : . [ m a t h . DG ] D ec Special bi-invariant linear connections on Liegroups and finite dimensional Poissonstructures
Saïd Benayadi and Mohamed Boucetta ∗ Abstract
Let G be a connected Lie group and g its Lie algebra. We denoteby ∇ the torsion free bi-invariant linear connection on G given by ∇ X Y = [ X, Y ] , for any left invariant vector fields X, Y . A Poissonstructure on g is a commutative and associative product on g for which ad u is a derivation, for any u ∈ g . A torsion free bi-invariant linearconnections on G which have the same curvature as ∇ is called spe-cial. We show that there is a bijection between the space of specialconnections on G and the space of Poisson structures on g . We com-pute the holonomy Lie algebra of a special connection and we showthat the Poisson structures associated to special connections whichhave the same holonomy Lie algebra as ∇ possess interesting proper-ties. Finally, we study Poisson structures on a Lie algebra and we givea large class of examples which gives, of course, a large class of specialconnections. Key words:
Lie groups, Lie algebras, bi-invariant linear connections, Pois-son algebras, symplectic Lie algebras, symplectic double extension, semi-symmetric linear connections.
MSC.
All vector spaces, algebras, etc. in this paper will be over a ground field K of characteristic 0. ∗ This research was conducted within the framework of Action concertée CNRST-CNRSProject SPM04/13. Poisson algebra is a finite dimensional Lie algebra ( g , [ , ]) endowed witha commutative and associative product ◦ such that, for any u, v, w ∈ g , [ u, v ◦ w ] = [ u, v ] ◦ w + v ◦ [ u, w ] . (1)An algebra (A , . ) is called Poisson admissible if (A , [ , ] , ◦ ) is a Poissonalgebra, where [ u, v ] = u.v − v.u and u ◦ v = 12 ( u.v + v.u ) . (2)Poisson algebras constitute an interesting topic in algebra and were studiedby many authors (see for instance [15, 21, 23]). This paper aims to givesome new insights on them based on an interesting geometric interpretationof these structures when the field is either R or C (see Theorem 2.1). Let uspresent briefly this geometric interpretation.Let G be a Lie group with g = T e G its Lie algebra. The linear connec-tion ∇ given by ∇ X Y = [ X, Y ] , where X, Y are left invariant vectorfields, is torsion free, bi-invariant, complete and its curvature K is givenby K ( X, Y ) = − ad [ X,Y ] . Moreover, ∇ K = 0 and the holonomy Lie al-gebra of ∇ at e is h = ad [ g , g ] . The main fact (see Section 2) is that thereis a bijection between the set of Poisson structures on g and the space of bi-invariant torsion free linear connections on G which have the same curvatureas ∇ . We call such connections special . Moreover, we show that any specialconnection is semi-symmetric, i.e., its curvature tensor K satisfies K.K = 0 (see Proposition 2.1). In general, the holonomy Lie algebra h of a bi-invariantlinear connection is difficult to compute, however, we show that, for a spe-cial connection, h contains h and can be easily computed (see Lemma 2.2).A special connection whose holonomy Lie algebra coincides with h will becalled strongly special . So, according to the bijection above, to any real Pois-son algebra corresponds a unique special connection on any associated Liegroup. Poisson algebras whose corresponding special connection is stronglyspecial are particularly interesting. We call such Poisson algebras strong .With this interpretation in mind, we devote Section 3 to the study of thegeneral properties of Poisson algebras and Poisson admissible algebras andwe give some general methods to build new Poisson algebras from old ones(see Theorem 3.2). We show that any symmetric Leibniz algebra is a strongPoisson admissible algebra and the curvature of the corresponding specialconnection is parallel (See Theorem 3.1). By using the geometric interpre-tation of Poisson structures, we get a large class of Lie groups which carrya bi-invariant connection ∇ (different from ∇ ) which has the same curva-ture and the same linear holonomy as ∇ and moreover the curvature of ∇
2s parallel. We get hence interesting examples of connections with paralleltorsion and curvature. Such connections were studied by Nomizu [22]. Re-call that symmetric Leibniz algebras constitute a subclass of Leibniz algebrasintroduced by Loday in [20]. At the end of Section 3, we show that there isno non trivial Poisson structure on a semi-simple Lie algebra (see Theorem3.3). This result generalizes a result by [15]. In Section 4, we show thatan associative algebra is Poisson admissible if and only if the underline Liealgebra is 2-nilpotent and we give a description of associative Poisson admis-sible algebras which permit to build many examples. Section 5 is devoted tothe study of symplectic Poisson algebras. It is well-known that if ( g , ω ) isa symplectic Lie algebra there is a product α a on g which is Lie-admissibleand left symmetric. When the Lie algebra is real, α a defines a left invariantflat torsion free linear connection ∇ a on any associated Lie group G . Byusing the general method to build a torsion free symplectic connection fromany torsion free connection introduced in [4], we get from ∇ a a left invari-ant torsion free connection ∇ s for which the left invariant symplectic formassociated to ω is parallel. To our knowledge this connection has never beenconsidered before. From ∇ s we get a product α s on g . We show that ( g , α a ) isPoisson admissible iff ( g , α s ) is Poisson admissible and this is equivalent to g is 2-nilpotent Lie algebra and [ad u , ad ∗ v ] = 0 for any u, v ∈ g where ad ∗ u is theadjoint of ad u with respect to ω . A symplectic Lie algebra satisfying theseconditions is called symplectic Poisson algebra . We show that the symplecticdouble extension process introduced in [12] permits the construction of allsymplectic Poisson algebras. Lie groups whose Lie algebras are symplecticPoisson possesses an important geometric property (see Theorem 5.2 and thefollowing remarks). In Section 6, we study the problem of metrizability ofspecial connections. Indeed, we consider a real Lie algebra ( g , [ , ] , h , i ) endowed with a nondegenerate symmetric bilinear metric. We denote by ℓ the Levi-Civita product associated to ( g , [ , ] , h , i ) . We show that if h , i is positive definite ( g , ℓ ) is Poisson admissible iff h , i is bi-invariant and inthis case the associated Poisson product ◦ is trivial. We give a descriptionof ( g , [ , ] , h , i ) for which ( g , ℓ ) is Poisson admissible in the case where [ g , g ] is nondegenerate and h , i has any signature. We give in this section an interesting geometric interpretation of Poissonstructures involving the theory of connections and holonomy algebras. This3heory is a fundamental topic in differential geometry and has its origin inthe work of Elie Cartan [5, 8]. For a detailed account of this theory, seeEhresmann [13], Chern [11], Lichnerowicz [18], Nomizu [22], and Kobayashi[16]. Let us recall some classical facts about linear connections and statesome formulas which will lead naturally to the desired interpretation.Given a linear connection on a smooth manifold M , we consider the covariantdifferentiation ∇ associated to it. Let T ∇ and K ∇ be, respectively, the torsionand curvature tensor fields on M with respect to ∇ : T ∇ ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] and K ∇ ( X, Y ) = [ ∇ X , ∇ Y ] − ∇ [ X,Y ] . For any closed curve τ at p ∈ M , the parallel displacement along τ is a lineartransformation of T p M , and the totality of these linear transformations forall closed curves forms the holonomy group H ( p ) . The restricted holonomygroup H ( p ) is the subgroup consisting of parallel displacements along allclosed curves which are homotopic to zero. Its Lie algebra is called holonomyLie algebra . On the other hand, consider linear endomorphisms of T p M ofthe form K ∇ ( X, Y ) , ( ∇ Z K ∇ )( X, Y ) , ( ∇ W ∇ Z K ∇ )( X, Y ) , . . . (all covari-ant derivatives), where X, Y, Z, W, . . . are arbitrary tangent vectors at p . All these linear endomorphisms span a subalgebra h ∇ p of the Lie algebraconsisting of all linear endomorphisms of T p M . We call it the infinitesimalholonomy Lie algebra . The Lie subgroup of GL( T p M, R ) generated by h ∇ p is the infinitesimal holonomy group at p . The main result is that if the in-finitesimal holonomy group has the same dimension at every point p of M (which is the case when M and ∇ are analytic), then the restricted holon-omy group is equal to the infinitesimal holonomy group at every point (see[22]). The linear connection ∇ will be called invariant under parallelism incase T ∇ and K ∇ are both parallel with respect to ∇ . The existence of alinear connection ∇ invariant under parallelism characterize (at lest locally)reductive homogeneous spaces (see [17]). If ∇ is invariant under parallelismthen h ∇ p = n X K ∇ ( u i , v i ) , u i , v i ∈ T p M o . (3)A vector field A is an infinitesimal ∇ -transformation if and only if for anycouple of vector fields X, Y , [ A, ∇ X Y ] = ∇ [ A,X ] Y + ∇ X [ A, Y ] . (4)On can see easily that this relation is equivalent to ∇ X,Y A + [ ∇ X , T ∇ A ] Y = K ∇ ( X, A ) Y, (5)where ∇ X,Y A = ∇ X ∇ Y A − ∇ ∇ X Y A . 4et ∇ be another linear connections on M . One knows that S = ∇ − ∇ is a tensor field of type (1 , . By using a terminology due to Kostant, wewill say that ∇ is rigid with respect to ∇ whenever S is parallel with respectto ∇ . In this case, we have the following formula (see [17] Lemma 2): K ∇ ( X, Y ) = K ∇ ( X, Y ) + [ S X , S Y ] + S T ∇ ( X,Y ) . (6)Let G be a connected Lie group, g = T e G its Lie algebra. For any u ∈ g wedenote by u + (resp. u − ) the left invariant (resp. the right invariant) vectorfield associated to u .It is obvious that G is a reductive homogeneous space and hence, accordingto a result of Kostant (See [17] Theorem 2), G admits a linear connectioninvariant under parallelism. In fact G admits many such connections and wewill use in this paper a special one, namely, the linear connection ∇ givenby ∇ u + v + = 12 [ u + , v + ] , for any u, v ∈ g . This connection is torsion free, invariant under parallelism,bi-invariant, complete and its curvature and holonomy Lie algebra are givenby K ∇ ( u + , v + ) w + = −
14 [[ u + , v + ] , w + ] , u, v, w ∈ g . (7) h ∇ e = ad [ g , g ] . (8)A linear connection ∇ on G is called bi-invariant if ∇ is invariant by left andright multiplication. The following lemma gives different characterizationsof bi-invariant linear connections on G . Lemma 2.1
Let ∇ be a linear connection on G . Then the following asser-tion are equivalent:1. ∇ is a bi-invariant linear connection.2. For any couple of left invariant (resp. right invariant) vector field X, Y , ∇ X Y is left invariant (resp. right invariant).3. For any couple of left invariant vector field X, Y , ∇ X Y is left invariantand the product α : g × g −→ g given by α ( u, v ) = ( ∇ u + v + )( e ) satisfies [ u, α ( v, w )] = α ([ u, v ] , w ) + α ( v, [ u, w ]) . (9)
4. For any couple of right invariant vector field
X, Y , ∇ X Y is right in-variant and the product β : g × g −→ g given by β ( u, v ) = ( ∇ u − v − )( e ) satisfies [ u, β ( v, w )] = β ([ u, v ] , w ) + β ( v, [ u, w ]) . . ∇ is left invariant and rigid with respect to ∇ .6. ∇ is right invariant and rigid with respect to ∇ . Proof.
Since G is connected, ∇ is bi-invariant if and only if, for any u ∈ g , u + and u − are infinitesimal ∇ -transformations, i.e., according to (4),for any couple of vector fields X, Y , [ u + , ∇ X Y ] = ∇ [ u + ,X ] Y + ∇ X [ u + , Y ] and [ u − , ∇ X Y ] = ∇ [ u − ,X ] Y + ∇ X [ u − , Y ] . Since G is a parallelizable by left invariant vector field and these vector fieldscommute with right invariant vector fields, these equations are equivalent to [ u + , ∇ v + w + ] = ∇ [ u + ,v + ] w + + ∇ v + [ u + , w + ] and [ u − , ∇ v + w + ] = 0 , v, w ∈ g . The group G is also prallelizable by right invariant vector field and hencethese equations are also equivalent to [ u − , ∇ v − w − ] = ∇ [ u − ,v − ] w − + ∇ v − [ u − , w − ] and [ u + , ∇ v − w − ] = 0 , v, w ∈ g . On the other hand, ∇ is left invariant and rigid with respect to ∇ is equiv-alent to [ u − , ∇ v + w + ] = 0 and ∇ u + ( ∇ − ∇ )( v + , w + ) = 0 . Or ∇ u + ( ∇ − ∇ )( v + , w + ) = ∇ u + (cid:0) ∇ v + w + − ∇ v + w + (cid:1) − (cid:16) ∇ ∇ u + v + w + − ∇ ∇ u + v + w + (cid:17) − (cid:0) ∇ v + ∇ u + w + − ∇ v + ∇ u + w + (cid:1) = 12 [ u + , ∇ v + w + ] −
14 [ u + , [ v + , w + ]] − ∇ [ u + ,v + ] w + −
14 [ w + , [ u + , v + ]] − ∇ v + [ u + , v + ] −
14 [ v + , [ w + , u + ]]= 12 [ u + , ∇ v + w + ] − ∇ [ u + ,v + ] w + − ∇ v + [ u + , v + ] . A similar computation holds when one replaces left invariant vector field byright ones. Now we can get the desired equivalences easily. (cid:3)
Remark 1
For any u, v ∈ g , u + ( e ) = u − ( e ) and [ v + , u − ] = 0 , so we get ( ∇ u + v + )( e ) = ( ∇ u − v + )( e ) = ( ∇ v + u − )( e ) = ( ∇ v − u − )( e ) . Thus α ( u, v ) = β ( v, u ) . ∇ be torsion free bi-invariant linear connection on G . As above, wedefine S = ∇ − ∇ . It is clear that S is bi-invariant and define a product ◦ on g . We have u ◦ v = α ( u, v ) −
12 [ u, v ] = 12 α ( u, v ) + 12 α ( v, u ) = 12 α ( u, v ) + 12 β ( u, v ) . This product is obviously commutative and, according to Lemma 2.1 3 and4, satisfies (1). Since ∇ is rigid with respect to ∇ , (6) holds and can bewritten for any u, v ∈ g , K ∇ ( u, v ) = K ∇ ( u, v ) + [ S u , S v ] . Thus K ∇ = K ∇ if and only if [ S u , S v ] = 0 for any u, v ∈ g , which isequivalent to ◦ is associative. Hence ◦ defines a Poisson structure on g if and only if ∇ and ∇ have the same curvature. So we get the desiredinterpretation. Theorem 2.1
Let G be a connected Lie group and g its Lie algebra. Thenthe following assertions hold:1. Let ∇ be a left invariant linear connection on g an let ◦ the product on g given by u ◦ v = ( ∇ u + v + )( e ) −
12 [ u, v ] . Then ( g , [ , ] , ◦ ) is a Poisson algebra if and only if ∇ is torsion free,bi-invariant and has the same curvature as ∇ .2. Let ◦ be a product on g such that ( g , [ , ] , ◦ ) is a Poisson algebra. Thenthe linear connection on G given by ∇ u + v + = 12 [ u + , v + ] + ( u ◦ v ) + is torsion free, bi-invariant and has the same curvature as ∇ . We call special a torsion free bi-invariant linear connection which has thesame curvature as ∇ .In Riemannian geometry there is a notion of semi-symmetric spaces whichis a direct generalization of locally symmetric spaces, namely, Riemannianmanifolds for which the curvature tensor K satisfies K.K = 0 , i.e., ∇ X ∇ Y K − ∇ Y ∇ X K − ∇ [ X,Y ] K = 0 , (10)7or any vector fields X, Y . Semi-Riemannian symmetric spaces were inves-tigated first by E. Cartan [9] and studied by many authors ([19, 10, 24]etc). More generally, we call a torsion free linear connection on a manifoldsemi-symmetric if its curvature tensor satisfies (10).
Proposition 2.1
Any special connection is semi-symmetric.
Proof.
Let ∇ be a special connection on a Lie group G . According toTheorem 2.1, its curvature K is given by K ( X, Y ) Z = −
14 [[
X, Y ] , Z ] , for any left invariant vector fields X, Y, Z . Now, it was shown in [24] pp. 532that the equation (10) is equivalent to [ K ( U, V ) , K ( X, Y )] = K ( K ( U, V ) X, Y ) + K ( X, K ( U, V ) Y ) , for any left invariant vector field X, Y, U, V . By replacing K in this relationby its expression above we get the desired result. (cid:3) Let ∇ be a left invariant linear connection on G . The holonomy Liealgebra is the smallest subalgebra h ∇ of End( g ) which contains all K ∇ ( u, v ) and satisfying [ ∇ u , h ∇ ] ⊂ h ∇ for any u ∈ g (see [22]). It is clear that itis difficult to compute h ∇ explicitly. However, the holonomy algebra of aspecial linear connection can be computed easily. Lemma 2.2
Let ∇ be a special connection on G . Then the holonomy Liealgebra of ∇ is given by h ∇ e = ad [ g , g ] + L [[ g , g ] , g ] = ad [ g , g ] + R [[ g , g ] , g ] , where L , R : g −→ End( g ) are given by L u = α ( u, . ) and R u = α ( ., u ) and α ( u, v ) = ( ∇ u + v + )( e ) . Proof.
This is a consequence of the following formulas which hold for anyspecial connection. We have, for any u, v ∈ g , [ad u , L u ] = L [ u,v ] , [ad u , R u ] = R [ u,v ] , [L u , L v ] = L [ u,v ] −
14 ad [ u,v ] , [R u , R v ] = − R [ u,v ] −
14 ad [ u,v ] . These formulas will be stated rigorously in the next section. (cid:3)
A special connection which has also the same holonomy Lie algebra as ∇ is called strongly special . 8 Poisson algebras and Poisson admissible al-gebras
In this section, we study Poisson algebras and Poisson admissible algebras inalgebraic point of view, having in mind the results of the previous section.Let ( g , [ , ]) be a finite dimensional Lie algebra and α : g × g −→ g aproduct on g . For any u ∈ g , we define L u , R u : g −→ g by L u v = α ( u, v ) and R u v = α ( v, u ) . Suppose that α is Lie-admissible, i.e., for any u, v ∈ g , α ( u, v ) − α ( v, u ) = [ u, v ] . Suppose also that α is bi-invariant, i.e., it satisfies (9). It is obvious that theproduct ◦ on g given by u ◦ v = α ( u, v ) −
12 [ u, v ] = 12 α ( u, v ) + 12 α ( v, u ) (11)is bi-invariant which is equivalent to [ S u , ad u ] = S [ u,v ] , ( ∗ ) for any u, v ∈ g where S u v = u ◦ v . If we denote by K α the curvature of α ,we get that K α ( u, v ) := [L u , L v ] − L [ u,v ] = [ S u + 12 ad u , S v + 12 ad v ] − S [ u,v ] −
12 ad [ u,v ]( ∗ ) = [ S u , S v ] −
14 ad [ u,v ] . This formula is the infinitesimal analog of (6). Thus we have proved thefollowing result.
Proposition 3.1
Let α : g × g −→ g a Lie-admissible product on g and ◦ given by (11) . Then ( g , [ , ] , ◦ ) is a Poisson algebra if and only if α isbi-invariant and, for any u, v ∈ g , K α ( u, v ) = −
14 ad [ u,v ] .
9n view of this proposition and Lemma 2.2, we can introduce this definition.
Definition 3.1
Let g be a finite dimensional Lie algebra.1. A product α on g is called quasi-canonical if it is Lie-admissible, bi-invariant and K α ( u, v ) = −
14 ad [ u,v ] .
2. The holonomy Lie algebra of a quasi-canonical product α on g is thesubalgebra of the Lie algebra End( g ) given by h α = ad [ g , g ] + L [ g , [ g , g ]] = ad [ g , g ] + R [ g , [ g , g ]] .
3. A quasi-canonical product α is called strongly quasi-canonical if h α =ad [ g , g ] . According to Proposition 3.1 there is a correspondence between the set ofPoisson products on a Lie algebra g and the set of quasi-canonical productson g . We call strong a Poisson product whose corresponding quasi-canonicalproduct is strongly quasi-canonical. The corresponding quasi-canonical prod-uct to the trivial Poisson product is α ( u, v ) = [ u, v ] .Let α be a quasi-canonical product on a Lie algebra g . Then (9) isequivalent to [ad u , L v ] = L [ u,v ] , (12)or [ad u , R v ] = R [ u,v ] , (13)for any u, v ∈ g . Since ad u = L u − R u , we get when we replace ad u in (12)that K α ( u, v ) = [R u , L v ] . (14)Note that the curvature of α vanishes if and only if α is associative. In thiscase, the Lie algebra g is 2-nilpotent because K α ( u, v ) = − ad [ u,v ] .Let us define the infinitesimal analog of the notion of a connection whichhas parallel curvature. A product α on a Lie algebra g has a parallel curvatureif, for any u, v, w ∈ g , ∇ K α ( u, v, w ) := [L u , K α ( v, w )] − K α ( α ( u, v ) , w ) − K α ( v, α ( u, w )) = 0 . Let us give now some properties of Poisson admissible algebras. Thedefinition of a Poisson admissible algebra was given in Section 1.10 roposition 3.2
Let (A , . ) be an algebra. Then the following conditions areequivalent:1. (A , . ) is a Poisson admissible algebra.2. For any u, v ∈ A , [L u , L v ] + [R u , R v ] + 2[L u , R v ] = 0 and K ( u, v ) = [R u , L v ] , where K ( u, v ) := [L u , L v ] − L [ u,v ] .
3. For any u, v ∈ A , [R u , R v ] + L [ u,v ] + 3[L u , R v ] = 0 .
4. For any u, v ∈ A , [L u , L v ] − R [ u,v ] + 3[R u , L v ] = 0 . Proof.
For any u, v ∈ A , put L u v = R v u = u.v , [ u, v ] = u.v − v.u , u ◦ v = ( u.v + v.u ) and K ( u, v ) = [L u , L v ] − L [ u,v ] .The algebra (A , . ) is Poisson admissible if and only if (A , [ , ]) is a Liealgebra and ” . ” is quasi-canonical with respect to (A , [ , ]) . This is equivalentto • K ( u, v ) w + K ( v, w ) u + K ( w, u ) v = 0 , (Bianchi identity) • K ( u, v ) (14) = [R u , L v ] , • [L u + R u , L v + R v ] = 0 . (The associativity of ◦ ),for any u, v ∈ A . Since from the second condition we deduce that [R u , L v ] = − [R v , L u ] (the flexibility), the conditions above are equivalent to • K ( u, v ) w + K ( v, w ) u + K ( w, u ) v = 0 , • [L u , L v ] − L [ u,v ] = [R u , L v ] , • [L u , L v ] + [R u , R v ] + 2[L u , R v ] = 0 .So 1. implies 2. Now its obvious that 2. implies 3.Let us show now that the third condition implies the first one. Note firstthat K ( u, v ) w = ass( v, u, w ) − ass( u, v, w ) ( ∗ ) ass( u, v, w ) = ( u.v ) .w − u. ( v.w ) = [R w , L u ]( v ) . ( ∗∗ ) Moreover, the third condition implies that ass( u, v, w ) = − ass( w, v, u ) andhence the product ” . ” is Lie-admissible if and only if ass( u, v, w ) + ass( v, w, u ) + ass( w, u, v ) = 0 . ( ∗ ∗ ∗ ) Now from 3. we deduce that u, v, w ) = R u ◦ R w ( v ) − R w ◦ R u ( v ) + R v ◦ R w ( u ) − R v ◦ R u ( w ) and we get easily that "." is Poisson-admissible and consequently K ( u, v ) =[R u , L v ] . Now this relation and 3. implies that [L u , L v ] + [R u , R v ] + 2[L u , R v ] = 0 . The condition 4. is equivalent to 3. is a consequence of the following remark.If (A , . ) is an algebra and ⋆ is the product given by u ⋆ v = − v.u then (A , . ) is Poisson admissible if and only if (A , ⋆ ) is Poisson admissible. In this casethe structures of Lie algebras of (A , . ) and (A , ⋆ ) coincident. (cid:3) Poisson admissible algebras is a subclass of Lie-admissible flexible alge-bras studied in [1]. Recall that an algebra is called flexible if its associatorsatisfies ass( u, v, w ) + ass( w, v, u ) = 0 , for any u, v, w . The third characterization of Poisson admissible algebras inProposition 3.2 appears first in [21]. Corollary 3.1
An associative algebra (A , . ) is Poisson admissible if and onlyif (A , [ , ]) is a 2-nilpotent Lie algebra. An algebra (A , . ) is called LR -algebra if, for any u, v ∈ A , [L u , L v ] = [R u , R v ] = 0 . It follows from Proposition 3.2 that a LR -algebra is Poisson admissible if andonly if it is associative.Let us introduce now an important class of strong Poisson admissiblealgebras. A left Leibniz algebra is an algebra (A , . ) such that for any u ∈ A ,the left multiplication L u is a derivation, i.e., for any v, w ∈ A , u. ( v.w ) = ( u.v ) .w + v. ( u.w ) . [L u , L v ] = L uv or [L u , R v ] = R uv . (15)A right Leibniz algebra is an algebra (A , . ) such that, for any u ∈ A , theright multiplication R u is a derivation, i.e., for any v, w ∈ A , ( v.w ) .u = ( v.u ) .w + v. ( w.u ) . This is equivalent to one of the two following relations [R u , R v ] = R vu or [R u , L v ] = L vu . (16)An algebra which is left and right Leibniz is called symmetric Leibnizalgebra . Leibniz algebras were introduced by Loday [20]. Many examples ofsymmetric Leibniz algebras can be found in [2]. By using (15) and (16), weget the following proposition. Proposition 3.3
The following assertions are equivalent:1. (A , . ) is a symmetric Leibniz algebra.2. For any u, v ∈ A , [L u , L v ] = L uv = − R uv .
3. For any u, v ∈ A , [R u , R v ] = R vu = − L vu . Any Lie algebra is a symmetric Leibniz algebra and any Leibniz algebrais Lie-admissible. However, the class of symmetric Leibniz algebras containsstrictly the class of Lie algebras. We can state now one of our main result.
Theorem 3.1
Let (A , . ) be a symmetric Leibniz algebra. Then (A , . ) is Pois-son admissible, ” . ” is strongly quasi-canonical on (A , [ , ]) and has parallelcurvature. Proof.
Put Q = [R u , R v ] + L [ u,v ] + 3[L u , R v ] . By using Proposition 3.3, we get Q = − L vu + L uv − L vu − uv = 0 . Now, according to Proposition 3.2 we get that (A , . ) is Poisson admissible.On the other hand, by using Proposition 3.3, we get that, for any u, v ∈ A , ad [ u,v ] = 2L [ u,v ] = 4L uv and K ( u, v ) = L vu ad [A , A] = L [A , A] . So the holonomy Lie algebra of ” . ” is ad [A , A] which prove that ” . ” strongly quasi-canonical. Finally, ∇ K ( u, v, w ) = [L u , L wv ] − L ( uw ) v − L w ( uv ) == L u ( wv ) − L ( uw ) v − L w ( uv ) = 0 , which completes the proof. (cid:3) By using the geometric interpretation of Poisson structures introduced inSection 2, we get the following interesting corollary.
Corollary 3.2
Let (A , . ) be a real symmetric Leibniz algebra which is not aLie algebra and G any connected Lie group associated to (A , [ , ]) . Then theleft invariant connection on G given by ∇ u + v + = ( u.v ) + is different from ∇ , strongly special and its curvature is parallel. Example 1
We give here an example of a 4-dimensional real symmetricLeibniz algebra for which we give the connected and simply connected Liegroup associated to the underlying Lie algebra and we give explicitly the twoconnections ∇ and ∇ appearing in the corollary above.We consider the symmetric Leibniz algebra product on R given in the canon-ical basis ( e , e , e , e ) by e .e = e , e .e = e , e .e = e , e .e = − e , e .e = − e . All the others products vanish. One can check easily by using Proposition 3.3that this product defines actually a symmetric Leibniz algebra. The underlyingLie algebra say g = R has its non-vanishing Lie brackets given by [ e , e ] = − e and [ e , e ] = − e . It is a 3-nilpotent Lie algebra. The associated connected and simply con-nected Lie group is G = R with the multiplication given by Campbell-Baker-Hausdorff formula xy = x + y + 12 [ x, y ] + 112 [ x, [ x, y ]] + 112 [ y, [ y, x ]] . This formula can be written xy = ( x + y , x + y , x + y − ( x y − x y ) , x + y − ( x y − x y )+ 13 x ( x y − x y ) + 13 y ( y x − y x )) . (17)14 ecall that for any vector u ∈ g , u + denotes the left invariant vector on G associated to u . A straightforward computation using (17) gives e +1 = ∂∂x + x ∂∂x + ( x − x x ) ∂∂x ,e +2 = ∂∂x − x ∂∂x + 13 x ∂∂x ,e +3 = ∂∂x − x ∂∂x , e +4 = ∂∂x , where ( x , x , x , x ) are the canonical coordinates of R . We consider thetwo torsion free linear connections on G defined by the formulas ∇ x + y + = 12 [ x + , y + ] and ∇ x + y + = ( x.y ) + . (18) The dot here is the symmetric Leibniz product. According to what abovethese two connections are bi-invariant, have the same curvature and the sameholonomy Lie algebra. Moreover, they both have parallel curvature. Let uscompute the Christoffel symbols of ∇ and ∇ in the canonical coordinates ( x , x , x , x ) . The computation is straightforward consisting of replacing x and y in (18) by e i , e j , i, j = 4 , . . . , . We get that the only non-vanishingChristoffel symbols are given by ∇ ∂∂x ∂∂x = − x ∂∂x and ∇ ∂∂x ∂∂x = 13 x ∂∂x , and ∇ ∂∂x ∂∂x = (1 − x ) ∂∂x and ∇ ∂∂x ∂∂x = 13 x ∂∂x . We can also compute the exponential maps associated to ∇ and ∇ and weget that exp : g −→ G is the identity, however exp : g −→ G is given by exp( a, b, c, d ) = ( a, b, c, d − a ) . Proposition 3.4
A left (right) Leibniz algebra is Poisson admissible if andonly if it is a symmetric Leibniz algebra.
Proof.
Suppose that (A , . ) is a left Leibniz algebra. According to Propo-sition 3.2, (A , . ) is Poisson admissible if and only if, for any u, v ∈ A , [L u , L v ] − R [ u,v ] + 3[R u , L v ] = 0 . L uv = 2R vu + R uv . On the otherhand, (15) implies that L uv = − L vu so we deduce that L uv = − R uv and wecan achieve the proof by using Proposition 3.3 and Theorem 3.1. (cid:3) The following proposition gives a tool to build many symmetric Leibnizalgebras from old ones.
Proposition 3.5
Let A be a symmetric Leibniz and U an associative LR -algebra then A ⊗ U endowed with the product ( u ⊗ a )( v ⊗ b ) = ( uv ) ⊗ ( ab ) is a symmetric Leibniz algebra. Proof.
It is a straightforward computation. (cid:3)
We can state now our second main result.
Theorem 3.2
Let (A , . ) be a Poisson admissible algebra and U an associa-tive LR -algebra. Then the product on A ⊗ U given by ( u ⊗ a ) ⋆ ( v ⊗ b ) = 12 [ u, v ] ⊗ ( ab + ba ) + 12 u.v ⊗ (3 ab + ba ) induces on A ⊗ U a Poisson admissible algebra structure. Moreover, if ” . ” isstrongly quasi-canonical on (A , [ , ]) then ⋆ is also strongly quasi-canonicalon (A ⊗ U , [ , ]) . Proof.
Note first that since U is an associative LR -algebra, for any a , a , a ∈ U and for any permutation σ of { , , } , a σ (1) a σ (2) a σ (3) = a a a .We will use Proposition 3.2 and show that, for any u, v ∈ A and a, b ∈ U , Q = [L u ⊗ a , L v ⊗ b ] − R [ u ⊗ a,v ⊗ b ] + 3[R u ⊗ a , L v ⊗ b ] = 0 . For any w ∈ A and c ∈ U , we have [L u ⊗ a , L v ⊗ b ]( w ⊗ c ) = ( u ⊗ a ) ⋆ (cid:18)
12 [ v, w ] ⊗ ( bc + cb ) + 12 v.w ⊗ (3 bc + cb ) (cid:19) − ( v ⊗ b ) ⋆ (cid:18)
12 [ u, w ] ⊗ ( ac + ca ) + 12 u.w ⊗ (3 ac + ca ) (cid:19) = ([ u, [ v, w ]] + 2 u. [ v, w ] + 2[ u, vw ] + 4 u ( vw )) ⊗ ( abc ) − ([ v, [ u, w ]] + 2 v. [ u, w ] + 2[ v, uw ] + 4 v ( uw )) ⊗ ( abc )= ([ u, [ v, w ]] + [ v, [ w, u ]] + 4[ u, v ] .w + 4[L u , L v ]( w )) ⊗ ( abc ) . [L u ⊗ a , L v ⊗ b ]( w ⊗ c ) = ([ u, [ v, w ]] + [ v, [ w, u ]] + 4[ u, v ] .w + 4[L u , L v ]( w )) ⊗ ( abc ) . A similar computation gives R [ u ⊗ a,v ⊗ b ] ( w ⊗ c ) = (4[ w, [ u, v ]] + 8 w. [ u, v ]) ⊗ ( abc ) , and [R u ⊗ a , L v ⊗ b ]( w ⊗ c ) = ([[ v, w ] , u ] − [ v, [ w, u ]]+2[[ v, u ] , w ]+4[R u , L v ]( w )) ⊗ ( abc ) . By using Jacobi identity and the relation [L u , L v ] − R [ u,v ] + 3[R u , L v ] = 0 , we get that Q = 0 and hence (A ⊗ U , . ) is a Poisson admissible algebra.On the other hand, a direct computation gives, for any u, v, w ∈ A and any a, b, c ∈ U , [[ u ⊗ a, v ⊗ b ] , w ⊗ c ] = 16[[ u, v ] , w ] ⊗ ( abc ) . This shows that [A ⊗ U , [A ⊗ U , A ⊗ U]] = [A , [A , A]] ⊗ U , and ad [ u ⊗ a,v ⊗ b ] = 16ad [ u,v ] ⊗ L ab . Moreover, one can check easily that for u ∈ [A , [A , A]] and a ∈ U , L u ⊗ a = (ad u + 2L u ) ⊗ L a . With all this formulas, one can show easily that if ” . ” is strongly quasi-canonical on (A , [ , ]) then ⋆ is also strongly quasi-canonical on (A ⊗ U , [ , ]) . (cid:3) Proposition 3.6
Let ( g , [ , ]) be a Lie algebra and ” . ” is a strongly quasi-canonical product on g . Then g = [ g , [ g , g ]] is two sided ideal of ( g , . ) , ( g , . ) is a symmetric Leibniz algebra and the sequence −→ ( g , . ) −→ ( g , . ) −→ ( g / g , . ) −→ is an exact sequence of Poisson admissible algebras, ( g / g , . ) is associativeand ( g / g , [ , ]) is 2-nilpotent. roof. Since ” . ” is strongly quasi-canonical then its holonomy Lie alge-bra is equal to ad [ g , g ] and hence L g ⊂ ad [ g , g ] and R g ⊂ ad [ g , g ] . Then for any u ∈ g and v ∈ g there exists w, t ∈ [ g , g ] such that L u = ad w and R u = ad t .So u.v ∈ g , v.u ∈ g and L u , R u are derivations of the restriction of ” . ” to g . We get that g is a two side ideal and ( g , . ) is a symmetric Leibnizalgebra. On the other hand, ( g / g , . ) is a Poisson algebra and ( g / g , [ , ]) is2-nilpotent so ( g / g , . ) is associative. (cid:3) In Proposition 23 of [15], it was proved that there is no non-trivial Pois-son structure on a simple complex Lie algebra. We finish this section bygeneralizing this result to any semi-simple Lie algebra over any field. Wewill show also that in a perfect Lie algebra the canonical product is the onlystrongly quasi-canonical product.
Theorem 3.3
1. Let g be a perfect Lie algebra, i.e., g = [ g , g ] . Then theproduct α ( u, v ) = [ u, v ] is the only strongly quasi-canonical producton g .2. Let g be a semi-simple Lie algebra. Then the product α ( u, v ) = [ u, v ] is the only quasi-canonical product on g . In particular, there is no nontrivial Poisson structure on g . Proof.
1. Suppose that ” . ” is a strongly quasi-canonical product on g and [ g , g ] = g . We have shown in Proposition 3.6 that in this case the restriction of ” . ” to [ g , [ g , g ]] is a Leibniz product. Or g = [ g , [ g , g ]] and hence ( g , . ) is a Leibniz algebra. Now from the relation L u.v = − R u.v and the factthat gg = g we deduce that u.v = [ u, v ] for any u, v ∈ g and hence ” . ” is the canonical product on g .2. Suppose that ” . ” is a quasi-canonical product on a semi-simple Liealgebra g , denote by L u and R u , respectively, the left and the rightmultiplication by u associated to ” . ” and put S u := L u − ad u . Notefirst that since g is semi-simple, g = [ g , g ] and hence, by using (12), weget that for any u ∈ g , tr( S u ) = 0 . Consider I = { u ∈ g , S u = 0 } . For any u ∈ g and any v ∈ I , we have from (12) that L [ u,v ] = [ad u , L v ] = 12 [ad u , ad v ] = 12 ad [ u,v ] , I is an ideal of ( g , [ , ]) . Let us show that I = g .Since g is semi-simple we have g = ⊕ pi =1 g i , where ( g i ) pi =1 is a family of simple ideals of g , [ g i , g i ] = g i , [ g i , g j ] = { } if i = j. We have for any i, j = 1 , . . . , p , g i . g i ⊂ g i and g i . g j = { } if i = j. Indeed, for any u, v ∈ g i and for j = i and w ∈ g j , we have [ w, u.v ] = [ w, u ] .v + u. [ w, v ] = 0 . By using a similar argument, we get that if i = j , u ∈ g i and v ∈ g j , u.v ∈ g i ⊕ g j . If u = [ a, b ] with a, b ∈ g i , we get u.v = [ a, b ] .v = [ a, b.v ] ∈ g i . Since [ g i , g i ] = g i we get that u.v ∈ g i and in a similar way u.v ∈ g j and hence u.v = 0 .Suppose by contradiction that I 6 = g . Since I is an ideal, eventuallyby changing the indexation of the family ( g i ) pi =1 , we can suppose thatthere exists ≤ r ≤ p such that g = I ⊕ J and J = ⊕ pi = r g i . For any u ∈ J , we denote by S u the restriction of S u to J . Theproduct on J given by u ◦ v = S u v is a Poisson product and hence itis commutative and associative. So, for any u ∈ J , and any n ∈ N ∗ , tr(( S u ) n ) = tr( S u n ) = tr( S u n ) = 0 , and hence S u is nilpotent. Since for any u, v ∈ J , [ S u , S v ] = 0 , wededuce by using Engel’s Theorem that there exists u ∈ J \ { } suchthat S u ( u ) = S u u = 0 . and hence S u = 0 . Since the restriction of S u to I vanishes, we deduce that S u = 0 and hence u ∈ I whichconstitutes a contradiction and achieves the proof. (cid:3) Associative Poisson admissible algebras
We have shown in Corollary 3.1 that an associative algebra (A , . ) is Poissonadmissible if and only if (A , [ , ]) is -nilpotent, i.e., for any u, v ∈ A , L [ u,v ] = R [ u,v ] . (19)An associative algebra satisfying this condition will be called associative Pois-son admissible algebra . This section is devoted to a description of such alge-bras.Let (A , . ) be an associative Poisson admissible algebra. We consider Z (A) = { u ∈ A , L u = R u } . Since A is associative, Z (A) is a commutative associative subalgebra of A .Put A = V ⊕ Z (A) , where V is a vectorial subspace of A . According to this splitting, we get that,for any z ∈ Z (A) and u, v ∈ V , z.u = u.z = P z ( u ) + Q u ( z ) and u.v = a ( u, v ) + b ( u, v ) . (20)These relations define two bilinear maps a : V × V −→ V , b : V × V −→ Z (A) ,and two linear maps Q : V −→ End( Z (A)) , P : Z (A) −→ End(V) . Thecondition (19) is equivalent to a symmetric and b ( u, v ) − b ( v, u ) = [ u, v ] .The associativity is equivalent to the following relations:1. P zz ′ = P z ◦ P z ′ , [Q u , L z ]( z ′ ) = Q P z ′ ( u ) ( z ) ,2. b (P z ( u ) , v ) + Q v ◦ Q u ( z ) = b ( u, P z ( v )) + Q u ◦ Q v ( z ) = Q a ( u,v )) ( z ) + z b ( u, v ) , P z ( a ( u, v )) = a ( u, P z ( v )) + P Q v ( z ) ( u ) , b ( a ( u, v ) , w ) − b ( u, a ( v, w )) = Q u ( b ( v, w )) − Q w ( b ( u, v )) , a ( a ( u, v ) , w ) − a ( u, a ( v, w )) = P b ( v,w ) ( u ) − P b ( u,v ) ( w ) . So, we have shown that the associative and commutative algebra Z (A) ,the vector space V , and P , Q , a , b as above satisfying the conditions 1-5describe entirely associative Poisson algebras. Proposition 4.1
Let ( g , [ , ]) be a 2-nilpotent Lie algebra. Then there is on g a quasi-canonical product different from the canonical one. roof. Put g = Z ( g ) ⊕ V and consider the product on g given, for any z, z ′ ∈ Z ( g ) , u, v ∈ V , by z.z ′ = u.z = z.u = 0 and u.v = s ( u, v ) + 12 [ u, v ] , where s is any non trivial symmetric bilinear map from V × V to Z ( g ) . Itis easy to check that this product is quasi-canonical and different from thecanonical one. (cid:3) In this section, we study an important class of Poisson algebras. To introducetheses algebras we recall some classical results on symplectic Lie groups andintroduce a new symplectic linear connection.Let ( G, Ω) be a symplectic Lie group, i.e., a Lie group G endowed with aleft invariant symplectic form Ω . It is well-known that the linear connectiongiven by the formula Ω( ∇ a u + v + , w + ) = − Ω( v + , [ u + , w + ]) , (21)where u, v, w ∈ g , defines a left invariant flat and torsion free connection ∇ a .Moreover, ∇ a Ω never vanishes unless G is abelian. So we can define a tensorfield N by the relation ∇ a u + Ω( v + , w + ) = Ω(N( u + , v + ) , w + ) . The linear connection given by ∇ s u + v + = ∇ a u + v + + 13 N( u + , v + ) + 13 N( v + , u + ) is left invariant torsion free and symplectic, i.e., ∇ s Ω = 0 . This constructionfollows a general scheme which permit to build symplectic connection fromany connection (see [4]). A straightforward computation gives that ∇ s canbe defined by the following formula Ω( ∇ s u + v + , w + ) = 13 Ω([ u + , v + ] , w + ) + 13 Ω([ u + , w + ] , v + ) . (22)This formula shows that on any symplectic Lie group there exists a canonicaltorsion free symplectic connection. 21et ( g , ω ) be the Lie algebra of G endowed with the value of Ω at e . Wedenote by α a and α s the product on g induced, respectively, by ∇ a and ∇ s .We have, for any u, v ∈ g , α a ( u, v ) = − ad ∗ u v and α s ( u, v ) = 13 (ad u v − ad ∗ u v ) , (23)where ad ∗ u is the adjoint of ad u with respect to ω .Conversely, given a symplectic Lie algebra, the formulas (23) defines on g two Lie-admissible products whose one is left symmetric and the other oneis symplectic. Let us see under which conditions these products are quasi-canonical. Proposition 5.1
Let ( g , ω ) be a symplectic Lie algebra and α a , α s the prod-uct given by (23) . Then the following assertions are equivalent:1. α a is quasi-canonical.2. α s is quasi-canonical.3. g is 2-nilpotent and, for any u, v ∈ g , [ad u , ad ∗ v ] = 0 .Moreover, if one of the conditions above holds then ( g , α a ) and ( g , α s ) areboth associative LR -algebras. Proof.
Not first that K α a = 0 and the left and right multiplicationsassociated to α a are given by L a u = − ad ∗ u and R a u = − ad ∗ u − ad u . The product α a is quasi-canonical if and only if, for any u, v ∈ g , K α a ( u, v ) = [R a u , L a u ] and [L a u + R a u , L a v + R a v ] = 0 which is obviously equivalent to ad [ u,v ] = [ad u , ad ∗ v ] = 0 . On the other hand, we have L s u = 13 (ad u − ad ∗ u ) and R s u = −
13 (2ad u + ad ∗ u ) , and hence 22 α s ( u, v ) = [L s u , L s v ] − L s[ u,v ] = 19 ([ad u , ad v ] − [ad u , ad ∗ v ] − [ad ∗ u , ad v ] + [ad ∗ u , ad ∗ v ]) −
13 ad [ u,v ] + 13 ad ∗ [ u,v ] , [R s u , L s v ] = −
19 (2[ad u , ad v ] − u , ad ∗ v ] + [ad ∗ u , ad v ] − [ad ∗ u , ad ∗ v ]) . Thus K α s ( u, v ) = [R s u , L s v ] if and only if [ad u , ad ∗ v ] = ad ∗ [ u,v ] . (24)Let us compute Q = [L s u + R s u , L s v + R s v ] . We have Q = 19 ([ad u , ad v ] + 2[ad u , ad ∗ v ] + 2[ad ∗ u , ad v ] + 4[ad ∗ u , ad ∗ v ]) . So we get that K α s ( u, v ) = [R s u , L s v ] and Q = 0 if and only if [ad u , ad ∗ v ] = ad [ u,v ] = 0 . (cid:3) A symplectic Poisson algebra is a 2-nilpotent symplectic Lie algebra ( g , ω ) satisfying, for any u, v ∈ g , [ad u , ad ∗ v ] = 0 . (25) Proposition 5.2
Let ( g , ω ) be 2-nilpotent symplectic Lie algebra which car-ries a bi-invariant pseudo-Euclidean product B . Then ( g , ω ) is a symplecticPoisson algebra. Proof.
We consider the isomorphism of g given by ω ( u, v ) = B ( Du, v ) . It is easy to check by using the fact that B is bi-invariant and ω is symplecticthat D is derivation of g and that, for any u ∈ g , ad ∗ u = − D − ◦ ad u ◦ D. Now, for any u, v ∈ g , [ad ∗ u , ad v ] = ad v ◦ D − ◦ ad u ◦ D − D − ◦ ad u ◦ D ◦ ad v . D ◦ ad v = ad v ◦ D + ad Dv and g is 2-nilpotent we get that D − ◦ ad u ◦ D ◦ ad v = 0 . On the other hand, D − [ g , g ] = [ g , g ] so we get since g is 2-nilpotent that ad v ◦ D − ◦ ad u ◦ D = 0 and finally, [ad ∗ u , ad v ] = 0 which show that ( g , ω ) isa symplectic Poisson algebra. (cid:3) Example 2
Let ( g , [ , ]) be a − nilpotent Lie algebra. Then g = V ⊕ Z ( g ) , where V is a vector subspace of g such that [ V , V ] ⊆ Z ( g ) . The endomorphism D of g defined by: D ( v ) = v and D ( z ) = 2 z, for all v ∈ V , z ∈ Z ( g ) , is an invertible derivation of g . Now , the vector space G = g ⊕ g ∗ endowed with the following product: [ u + α, v + β ] = [ u, v ] + α ◦ ad v − β ◦ ad u , for all u, v ∈ g , α, β ∈ g ∗ , is a − nilpotent Lie algebra. Moreover, the bilinear form B : G × G → K defined by: B ( u + α, v + β ) = α ( v ) + β ( u ) , for all u, v ∈ g , α, β ∈ g ∗ , is non-degenerate, bi-invariant and symmetric. Then ( G , B ) is a − nilpotentquadratic Lie algebra. An easy computation shows that the endomorphism δ of G defined by: δ ( u ) = D ( u ) and δ ( α ) = − α ◦ D, for all u ∈ g , α ∈ g ∗ , is an invertible derivation of G which is skew-symmetric with respect to B .Consequently, the bilinear form defined by: ω ( X, Y ) = B ( δ ( X ) , Y ) , for all X, Y ∈ G , is a symplectic structure on G . Finally, ( G , ω ) is symplectic Poisson algebra. Let us give now the inductive description of symplectic Poisson algebras.Let ( g , [ , ] g , ω ) be a symplectic Poisson algebra. Since g is nilpotent Liealgebra, according to [12], ( g , [ , ] g , ω ) is the symplectic double extensionof a symplectic Lie algebra ( h , [ , ] h , ω ) of dimension dim g − by the one-dimensional Lie algebra by means an element ( D, z ) of Der( h ) × h . Thismeans that g = K e ⊕ h ⊕ K d and 24. for any a, b ∈ h , [ a, b ] g = [ a, b ] h + ω (( D + D ∗ )( a ) , b ) e, [ d, d ] g = 0 , [ a, d ] g = D ( a ) + ω ( z, a ) e, [ e, g ] g = { } , where D ∗ the adjoint of D with respect to ω, ω | h × h = ω, ω ( e, d ) = 1 , ω ( e, h ) = ω ( d, h ) = { } . The fact that ( g , ω ) is symplectic Poisson algebra is equivalent to ad [ u,v ] = [ad u , ad ∗ v ] = 0 , for any u, v ∈ g .The first condition which means that g is 2-nilpotent Lie algebra is equiv-alent to: • h is a 2-nilpotent Lie algebra, • D ( h ) ⊂ Z ( h ) , • D | [ h , h ] h = D ∗| [ h , h ] h = 0 , ω ([ h , h ] h , z ) = 0 , • D = D ∗ ◦ D = 0 and D ∗ ( z ) = 0 .Let us compute ad ∗ u for any u ∈ g . A straightforward computation gives, forany a, b ∈ h , ad ∗ a b = ad h ∗ a b + ω ( b, D ( a )) e, ad ∗ a d = − ( D + D ∗ )( a ) + ω ( a, z ) e, ad ∗ a e = ad ∗ d e = 0 , ad ∗ d d = − z, ad ∗ d a = − D ∗ ( a ) . So [ad a , ad ∗ b ]( c ) = [ a, ad h ∗ b c ] − ad ∗ b [ a, c ] = [ a, ad h ∗ b c ] − ad h ∗ b [ a, c ] h + ω ([ a, c ] h , D ( b )) e = [ad h a , ad h ∗ b ]( c ) + ω (( D + D ∗ )( a ) , ad h ∗ b c ) e + ω ([ a, c ] h , D ( b )) e, [ad a , ad ∗ b ]( d ) = − [ a, ( D + D ∗ )( b )] − ad ∗ b D ( a )= − [ a, ( D + D ∗ )( b )] h − ad h ∗ b D ( a ) − ω (( D + D ∗ )( a ) , ( D + D ∗ )( b )) e − ω ( D ( a ) , D ( b )) e, [ad a , ad ∗ d ]( b ) = − [ a, D ∗ ( b )] + D ∗ ([ a, b ] h ) = − [ a, D ∗ ( b )] h − ω (( D + D ∗ )( a ) , D ∗ ( b )) e + D ∗ ([ a, b ] h ) , [ad a , ad ∗ d ]( d ) = − [ a, z ] + D ∗ ◦ D ( a )= − [ a, z ] h − ω (( D + D ∗ )( a ) , z ) e + D ∗ ◦ D ( a ) , [ad d , ad ∗ d ]( a ) = − [ d, D ∗ ( a )] − ad ∗ d [ a, d ] = D ◦ D ∗ ( a ) − ω ( z, D ∗ ( a )) e + D ∗ ◦ D ( a ) , [ad d , ad ∗ d ]( d ) = − D ( z ) . ( g , ω ) is symplectic Poisson algebra if andonly if • ( h , ω ) is a symplectic Poisson algebra, • D ( h ) ⊂ Z ( h ) , D ∗ ( h ) ⊂ Z ( h ) , • D = D ∗ ◦ D = D ◦ D ∗ = 0 , D ∗ ( z ) = D ( z ) = 0 and z ∈ Z ( h ) .An element ( D, z ) of Der( h ) × h which verifies the conditions above willbe called admissible .To summarize, we have proved the following theorem. Theorem 5.1
Let ( g , ω ) be a symplectic Lie algebra. Then ( g , ω ) is a sym-plectic Poisson algebra if and only if it is a symplectic double extension of asymplectic Poisson algebra ( h , ω ) of dimension dim g − by the one dimen-sional Lie algebra by means of an admissible element ( D, z ) ∈ Der( h ) × h . There is only one 2-dimensional symplectic Poisson algebra, namely thetwo 2-dimensional abelian Lie algebra h endowed with a symplectic form ω .There exists a basis B = { e , e } of h such that ω ( e , e ) = 1 . An element ( D, z ) ∈ Der( h ) × h is admissible if and only if z is any element of h andthe matrix of D in the basis B has one of the following forms (cid:18) a (cid:19) , (cid:18) a (cid:19) , (cid:18) a b − a b − a (cid:19) , b = 0 . So we get all four dimensional symplectic Poisson algebras.
Proposition 5.3
Let g be a 4-dimensional Lie algebra. Then g is a sym-plectic Poisson algebra iff it is isomorphic to one of the following symplecticLie algebras:1. span { e, e , e , d } with the non vanishing brackets [ e , d ] = − z e, [ e , d ] = − ae + z e, and the symplectic form satisfying ω ( e, d ) = ω ( e , e ) = 1 , ω ( e, e ) = ω ( e, e ) = ω ( d, e ) = ω ( d, e ) = 0 . span { e, e , e , d } with the non vanishing brackets [ e , d ] = ae − a b e − z e, [ e , d ] = be − ae + z e, and the symplectic form satisfying ω ( e, d ) = ω ( e , e ) = 1 , ω ( e, e ) = ω ( e, e ) = ω ( d, e ) = ω ( d, e ) = 0 .
26e finish this section by giving an important geometric property of realsymplectic Poisson algebras.Let ( g , ω ) be a non abelian real symplectic Poisson algebra and G a connectedLie group having g as its Lie algebra. The symplectic form ω defines on G asymplectic left invariant form Ω . Consider the two linear connections ∇ a and ∇ s defined on G by (21)-(22). These two connections are bi-invariant, flat,complete and ∇ s Ω = 0 . It was shown in [3] that Ω is polynomial of degreeat most dim G − in any affine coordinates chart associated to ∇ a . Thefollowing result gives a more accurate statement on the polynomial nature of Ω . Theorem 5.2
With the hypothesis and the notations above we have ( ∇ a ) Ω = 0 . In particular, Ω is polynomial of degree at least one and at most in anyaffine coordinates chart associated to ∇ a . Moreover, if the restriction of ω to [ g , g ] does not vanish then the degree is 2. Proof.
For any u, v, x, y ∈ g , an easy computation gives ∇ a u + Ω( x + , y + ) = Ω( u + , [ x + , y + ]) , and hence ∇ a u + ∇ a v + Ω( x + , y + ) = Ω([ ∇ a u + x + , y + ] + [ x + , ∇ a u + y + ] , v + ) . Now since ∇ a is bi-invariant then [ ∇ a u + x + , y + ] + [ x + , ∇ a u + y + ] = ∇ a[ u + ,y + ] x + + ∇ a[ x + ,u + ] y + + 2 ∇ a u + [ x + , y + ] . By using (21) and the fact that g is 2-nilpotent, we get ∇ a u + ∇ a v + Ω( x + , y + ) = 2Ω([ x + , y + ] , [ u + , v + ]) . By using the same arguments as above one can get easily that ( ∇ a ) Ω = 0 . The properties of the degree of Ω are an immediate consequence of formulasabove. (cid:3) It was proved in [14] that a compact affine manifold M has a polynomialRiemannian metric iff M is finitely covered by a complete affine nilmanifold.An affine nilmanifold is of the form Γ /N where N is a simply-connectednilpotent Lie group with a left invariant affine structure and Γ is a discrete27ubgroup of N . According to the results of this section, if G is the simply-connected Lie group associated to a non abelian symplectic Poisson Lie al-gebra and Γ is a co-compact discrete subgroup of G then Γ /G is a compactnilmanifold which carries two affine structures and a symplectic form whichis parallel for one affine structure and polynomial of degree at least 1 and atmost 2 for the other one. It is natural to ask if there is a symplectic analogof Goldman’s Theorem in [14]. In this section we study the problem of metrizability of special connectionson Lie groups. Given a connected Lie group G with ∇ a special connection,does exist on G a left invariant pseudo-Riemannian metric whose associatedLevi-Civita connection is ∇ ? Remark that if such a metric exists and it isbi-invariant then ∇ coincides with ∇ . The following proposition gives ananswer to this question when the metric is Riemannian. Proposition 6.1
Let g be a real Lie algebra and h , i an Euclidean producton g such that the associated Levi-Civita product is quasi-canonical. Then h , i is bi-invariant and hence the Levi-Civita product coincides with thecanonical product. Proof.
We have g = [ g , g ] ⊕ [ g , g ] ⊥ . Since for any u, v ∈ g , K ( u, v ) = − ad [ u,v ] and K ( u, v ) is skew-symmetric,we deduce that, for any w ∈ [ g , g ] , ad w is skew-symmetric. From this remarkand the relation h u.v, w i = h [ u, v ] , w i + h [ w, v ] , u i + h [ w, u ] , v i , one can deduce easily that, for any u, v ∈ [ g , g ] and any x, y ∈ [ g , g ] ⊥ , u.v = 12 [ u, v ] and x.y = 12 [ x, y ] . Let u ∈ [ g , g ] and v ∈ [ g , g ] ⊥ , since ad u is skew-symmetric, we get for any w ∈ g , h [ u, v ] , w i = −h v, [ u, w ] i = 0 , and hence u.v = v.u . Moreover, for any x ∈ [ g , g ] , h u.v, x i = −h v, u.x i = 12 h v, [ x, u ] i = 0 . u.v = v.u ∈ [ g , g ] ⊥ . Now h u.v, u.v i = h [ u, v ] , u.v i + h [ u.v, u ] , v i + h [ u.v, v ] , u i = 0 , since [ u, v ] = 0 and [ v, u.v ] = [ v, u ] .v + u. [ v, v ] = 0 . Thus u.v = 0 whichcompletes the proof. (cid:3) The proposition above is not true in general when the h , i is not positivedefinite. We give now a description of all real Lie algebras endowed withan pseudo-Euclidean product such that the associated Levi-Civita productis quasi-canonical and the derived ideal is non-degenerate.Consider ( h , h , i ) a Lie algebra endowed with a bi-invariant pseudo-Euclidean product. Let ( V, B ) be a vector space with a nondegeneratesymmetric bilinear form. We can split V = V ⊕ U ⊕ V such that therestriction of B to U is positive definite and the map V × V −→ R , ( u, v ) −→ B ( u, v ) is non-degenerate. Finally, consider any bilinear skew-symmetric map γ : V × V −→ Z ( h ) . We consider now g = h ⊕ V endowedwith h , i = h , i + B and the bracket for which V ⊕ U ⊂ Z ( g ) , [ h , V ] = 0 ,the restriction to h coincides with the initial bracket and for any u, v ∈ V , [ u, v ] = γ ( u, v ) . Then one can check easily that the Levi-Civita product of h , i is quasi-canonical and h , i is not bi-invariant. By a direct computationwe can see easily that the curvature tensor is parallel which gives examplesof locally symmetric pseudo-Riemannian spaces. References [1] G. M. Benkart and J. M. Osborn, Flexible Lie-admissible algebras, Jour-nal of Algebra, 71, No.1, (1981), 11-31.[2] S. Benayadi and S. Hidri, Quadratic Leibniz algebras, preprint, 2012.[3] M. Boucetta and A. Medina, Polynomial Poisson structures on someaffine solvmanifolds, Journal of Symplectic Geometry, Volume 9, Num-ber 3, 1-15, 2011.[4] P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley, L. Schwachhöfer, Sym-plectic connections, International Journal of Geometric Methods inModern Physics, Vol. 3, No. 3 (2006) 375-420295] E. Cartan, Les groupes d’holonomie des espaces généralisés, Acta Math.48 (1926), l-42.[6] E. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. Soc.Math. France 54 (1926), 214-264; 55 (1927), 114-134.[7] E. Cartan, La theorie des groupes finis et continus et l’analysis situs,Mém. Sci. Math. XL11 (1930).[8] E. Cartan, Sur les domaines bornés de l’espace de n -variables complexes,Ann. Math. Sem. Univ. Hamburg 11 (1935), 116-162.[9] E. Cartan, Leçons sur la géométrie des espaces de Riemann, 2nd ed.,Paris, 1946.[10] R. Couty, Sur les transformations des variétés riemanniennes et Kähle-riennes, Ann. Inst. Fourier (Grenoble) 9 (1959) 147-248.[11] S. S. Chern, Topics in Differential Geometry, Inst. Advanced Study,Princeton, New Jersey, 1951.[12] J.-M. Dardié and A. Medina, Double extension symplectique d’ungroupe de Lie symplectique, Adv. Math., 117, No.2 (1996), 208–227.[13] C. Ehresmann, Les connexions infinitésimales dans un espace fibré dif-férentiable, Colloq. Topol. Bruxelles (1950), 29-50.[14] Goldman, William M., A Generalization of Bieberbach’s Theorem, In-vent. Math. (1981), 1-11.[15] M. Goze and E. Remm, Poisson algebras in terms of non-associativealgebras, Journal of Algebra, 320, No.1, (2008), 294–317.[16] S. Kobayashi, Theory of connections, Ann. Mat. Pura Appl. Ser. 4, 43(1957), 119-194.[17] B. Kostant, A characterization of invariant affine connections, NagoyaMath. J. 16 (1960), 35-50.[18] A. Lichnerowich, Théorie globale des connexions et des groupesd’holonomie, Edizioni Cremonese, Roma, 1957.[19] A. Lichnerowich, Courbure, nombres de Betti et espaces symétriques,Proc. Internat. Congress Math. (Cambridge 1950), Amer. Math. Soc.Vol. II, 1952, 216-223. 3020] J. L. Loday, Une version non commutative des algèbres de Lie: les al-gèbres de Leibniz, Enseign. Math. 39 (1993), 269-293.[21] M. Markl and E. Remm, Algebras with one operation including Poissonand other Lie-admissible algebras, Journal of Algebra, 299, No.1, (2006),171–189.[22] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer-ican Journal of Mathematics, 76 (1954), pp. 33-65.[23] I.S. Shestakov, Quantization of Poisson superalgebras and speciality ofJordan Poisson superalgebras, Algebra and Logic, 32, N 5 (1993).[24] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R ( X, Y ) .R = 0= 0