aa r X i v : . [ m a t h . R A ] D ec K ¨AHLER–POISSON ALGEBRAS
JOAKIM ARNLIND AND AHMED AL-SHUJARY
Abstract.
We introduce K¨ahler–Poisson algebras as analogues of algebras ofsmooth functions on K¨ahler manifolds, and prove that they share several prop-erties with their classical counterparts on an algebraic level. For instance, themodule of inner derivations of a K¨ahler–Poisson algebra is a finitely generatedprojective module, and allows for a unique metric and torsion-free connectionwhose curvature enjoys all the classical symmetries. Moreover, starting froma large class of Poisson algebras, we show that every algebra has an associatedK¨ahler–Poisson algebra constructed as a localization. At the end, detailedexamples are provided in order to illustrate the novel concepts. Introduction
Poisson manifolds and their geometry have been of great interest over the lastdecades. Besides from being important from a purely mathematical point of view,they are also fundamental to areas in mathematical and theoretical physics. Manyauthors have studied the geometric and algebraic properties of symplectic and Pois-son manifolds together in relation to concepts such as connections, local structureand cohomology (see e.g. [Lic77, Wei83, Bry88, Hue90]). Moreover, there is a welldeveloped field of deformations of Poisson structures, perhaps most famous throughKontsevich’s result on the existence of formal deformations [Kon03]. The ring ofsmooth functions on a Poisson manifold is a Poisson algebra and it seems quite nat-ural to ask to what extent geometric properties and concepts may be introduced inan arbitrary Poisson algebra, without making reference to an underlying manifold.The methods of algebraic geometry can readily be extended to Poisson algebras(see e.g. [Ber79]); however, this will not be directly relevant to us as we shallstart by focusing on metric aspects. Our work is mainly motivated by the results in[AHH12, AH14], where it is shown that one may reformulate the Riemannian geom-etry of an embedded K¨ahler manifold M entirely in terms of the Poisson structureon the algebra smooth functions of M . Let us also mention that the starting pointof our approach is quite similar to that of [Hue90] (although metric aspects werenot considered there).In this note, we show that any Poisson algebra, fulfilling an “almost K¨ahler con-dition”, enjoys many properties similar to those of the algebra of smooth functionson an almost K¨ahler manifold, opening up for a more metric treatment of Poissonalgebras. Such algebras will be called “K¨ahler–Poisson algebras”, and we showthat one may associate a K¨ahler–Poisson algebra to every algebra in a large classof Poisson algebras. In particular, we prove the existence of a unique Levi-Civitaconnection on the module generated by the inner derivations, and show that thecurvature operator has all the classical symmetries. As our approach is quite closeto the theory of Lie-Rinehart algebras, we start by introducing metric Lie-Rinehart algebras and recall a few results on the Levi-Civita connection and the correspond-ing curvature.In physics, the dynamics of quantum systems are found by using a correspon-dence between Poisson brackets of functions on the classical manifold, and the com-mutator of operators in the quantum system. Thus, understanding how propertiesof the underlying manifold may be expressed in Poisson algebraic terms enablesboth interpretation and definition of quantum mechanical quantities. For instance,this has been used in the context of matrix models to identify emergent geometry(cf. [BS10, AHH12]).Let us briefly outline the contents of the paper. In Section 2 we recall a fewthe results from [AH14], in order to motivate and understand the introduction of aK¨ahler type condition for Poisson algebras, and Section 3 explains how the theoryof Lie-Rinehart algebras can be extended to include metric aspects. In Section 4,we define K¨ahler–Poisson algebras and investigate their basic properties as well asshowing that one may associate a K¨ahler–Poisson algebra to an arbitrary Poissonalgebra in a large class of algebras. In Section 5 we derive a compact formula for theLevi-Civita connection as well as introducing Ricci and scalar curvature. Section 6presents a number of examples together with a few detailed computations. Remark . We have become aware of the fact that the terminology
K¨ahler–Poisson structure (resp.
K¨ahler–Poisson manifold ) is used for certain Poissonstructures on a complex manifold where the Poisson bivector is of type (1 ,
1) (seee.g. [Kar02]), but we hope that this will not be a source of confusion for the reader.2.
Poisson algebraic formulation of almost K¨ahler manifolds
In [AH14] it was shown that the geometry of embedded almost K¨ahler manifoldscan be reformulated entirely in the Poisson algebra of smooth functions. As we shalldevelop an algebraic analogue of this fact, let us briefly recall the main construction.Let (Σ , ω ) denote a n -dimensional symplectic manifold and let g be a metric onΣ. Furthermore, let us assume that x : (Σ , g ) → ( R m , ¯ g ) is an isometric embeddingof Σ into R m (with the metric ¯ g ), and write p → x ( p ) = (cid:0) x ( p ) , x ( p ) , . . . , x m ( p ) (cid:1) . The results in [AH14] state that the Riemannian geometry of Σ may be formulatedin terms of the Poisson algebra generated by the embedding coordinates x , . . . , x m .These results hold true as long as there exists a non-zero function γ ∈ C ∞ (Σ) suchthat γ g ab = θ ap θ bq g pq (2.1)where θ ab and g ab denote the components of the Poisson bivector and the metricin local coordinates { u a } na =1 , respectively. If (Σ , ω, g ) is an almost K¨ahler manifoldthen it follows from the compatibility condition ω ( X, Y ) = g ( X, J ( Y )) (where J denotes the almost complex structure on Σ) that relation (2.1) holds with γ = 1.In local coordinates, the isometric embedding is characterized by g ab = ¯ g ij (cid:0) ∂ a x i (cid:1)(cid:0) ∂ b x j (cid:1) , and the Poisson bracket is computed as { f, h } = θ ab ( ∂ a f )( ∂ b h ) . ¨AHLER–POISSON ALGEBRAS 3 Note that in the above and following formulas, indices i, j, k, . . . run from 1 to m and indices a, b, c, . . . run from 1 to n .Defining D : T p R m → T p R m as D ( X ) ≡ D ij X j ∂ i = 1 γ { x i , x k } ¯ g kl { x j , x l } ¯ g jm X m ∂ i for X = X i ∂ i ∈ T p R m , one computes D ( X ) i = 1 γ θ ab ( ∂ a x i )( ∂ b x k )¯ g kl θ pq ( ∂ p x j )( ∂ q x l )¯ g jm X m = 1 γ θ ab θ pq g bq ( ∂ a x i )( ∂ p x j )¯ g jm X m = g ap ( ∂ a x i )( ∂ p x j )¯ g jm X m , by using (2.1). Hence, the map D is identified as the orthogonal projection onto T p Σ, seen as a subspace of T p R m . Having the projection operator at hand, onemay directly proceed to develop the theory of submanifolds. For instance, theLevi-Civita connection ∇ on Σ is given by ∇ X Y = D (cid:0) ¯ ∇ X Y (cid:1) where X, Y ∈ Γ( T Σ) and ¯ ∇ is the Levi-Civita connection on ( R m , ¯ g ). In theparticular case (but generically applicable, by Nash’s theorem [Nas56]) when ¯ g isthe Euclidean metric, the above formula reduces to ∇ X Y i = 1 γ m X i,j,k,l,n =1 { x i , x k }{ x j , x k } X l { x l , x n }{ Y j , x n } . As we intend to develop an analogous theory for Poisson algebras, without anyreference to a manifold, we would like to reformulate (2.1) in terms of Poisson alge-braic expressions. Using that g ab = ¯ g ij ( ∂ a x i )( ∂ b x j ) and { x i , x k } = θ ab ( ∂ a x i )( ∂ b x j ),one derives γ g ab = θ ap θ bq g pq ⇒ γ δ ac = θ ap θ bq g pq g bc ⇒ γ θ ar = θ ap θ bq g pq g bc θ cr ⇒ γ { x i , x j } = ( ∂ a x i )( ∂ r x j ) θ ap θ bq θ cr ¯ g kl ( ∂ p x k )( ∂ q x l )¯ g mn ( ∂ b x m )( ∂ c x n ) ⇒ γ { x i , x j } = −{ x i , x k } ¯ g kl { x l , x n } ¯ g nm { x m , x j } which is equivalent to the statement that γ { f, h } = −{ f, x i } ¯ g ij { x j , x k } ¯ g kl { x l , h } (2.2)for all f, h ∈ C ∞ (Σ). Given γ , ¯ g ij and x , . . . , x m , the above equation makessense in an arbitrary Poisson algebra. The main purpose of this paper is to studyalgebras which satisfy such a relation.3. Metric Lie-Rinehart algebras
The idea of modeling the algebraic structures of differential geometry in a commu-tative algebra is quite old. We shall follow a pedestrian approach, were we assumethat a (commutative) algebra A is given (corresponding to the algebra of functions),together with an A -module g (corresponding to the module of vector fields) whichis also a Lie algebra and has an action on A as derivations. Under appropriate as-sumptions on the ingoing objects, such systems has been studied by many authorsover the years, see e.g [Her53, Koz60, Pal61, Rin63, Nel67, Hue90]. Our starting JOAKIM ARNLIND AND AHMED AL-SHUJARY point is the definition given by G. Rinehart [Rin63]. In the following, we let thefield K denote either R or C . Definition 3.1 (Lie-Rinehart algebra) . Let A be a commutative K -algebra and let g be an A -module which is also a Lie algebra over K . Given a map ω : g → Der( A ),the pair ( A , g ) is called a Lie-Rinehart algebra if ω ( aα )( b ) = a (cid:0) ω ( α )( b ) (cid:1) (3.1) [ α, aβ ] = a [ α, β ] + (cid:0) ω ( α )( a ) (cid:1) β, (3.2)for α, β ∈ g and a, b ∈ A . (In most cases, we will leave out ω and write α ( a ) insteadof ω ( α )( a ).)Let us point out some immediate examples of Lie-Rinehart algebras. Example 3.2.
Let A be an algebra and let g = Der( A ) be the A -module of deriva-tions of A . It is easy to check that Der( A ) is a Lie algebra with respect to compo-sition of derivations, i.e. [ α, β ]( a ) = α ( β ( a )) − β ( α ( a )) . The pair ( A , Der( A )) is a Lie-Rinehart algebra with respect to the action of elementsof Der( A ) as derivations. Example 3.3.
Let A = C ∞ ( M ) be the algebra (over R ) of smooth functions ona manifold M , and let g = X ( A ) be the A -module of vector fields on M . Withrespect to the standard action of a vector field as a derivation of C ∞ ( M ) , the pair ( C ∞ ( M ) , X ( A )) is a Lie-Rinehart algebra. Morphisms of Lie-Rinehart algebras are defined as follows.
Definition 3.4.
Let ( A , g ) and ( A , g ) be Lie-Rinehart algebras. A morphism of Lie-Rinehart algebras is a pair of maps ( φ, ψ ), with φ : A → A an algebrahomomorphism and ψ : g → g a Lie algebra homomorphism, such that ψ ( aα ) = φ ( a ) ψ ( α ) and φ (cid:0) α ( a ) (cid:1) = ψ ( α ) (cid:0) φ ( a ) (cid:1) , for all a ∈ A and α ∈ g .A lot of attention has been given to the cohomology of the Chevalley–Eilenbergcomplex consisting of alternating A -multilinear maps with values in a module M . Namely, defining C k ( g , M ) to be the A -module of alternating maps from g k to an ( A , g )-module M , on introduces the standard differential d : C k ( g , M ) → C k +1 ( g , M ) as dτ ( α , . . . , α k +1 ) = k +1 X i =1 ( − i +1 α i (cid:0) τ ( α , . . . , ˆ α i , . . . , α k +1 ) (cid:1) + k +1 X i Let ( A , g ) be a Lie-Rinehart algebra and let M be an A -module.An A -bilinear form g : M × M → A is called a metric on M if it holds that(1) g ( m , m ) = g ( m , m ) for all m , m ∈ M ,(2) the map ˆ g : M → M ∗ , given by (cid:0) ˆ g ( m ) (cid:1) ( m ) = g ( m , m ), is an A -moduleisomorphism,where M ∗ denotes the dual of M . We shall often refer to property (2) as the metricbeing non-degenerate . Definition 3.6. A metric Lie-Rinehart algebra ( A , g , g ) is a Lie-Rinehart algebra( A , g ) together with a metric g : g × g → A .Let us introduce morphisms of metric Lie-Rinehart algebras as morphisms of Lie-Rinehart algebras that preserve the metric. Definition 3.7. Let ( A , g , g ) and ( A , g , g ) be metric Lie-Rinehart algebras.A morphism of metric Lie-Rinehart algebras is a morphism of Lie-Rinehart algebras( φ, ψ ) : ( A , g ) → ( A , g ) such that φ (cid:0) g ( α, β ) (cid:1) = g (cid:0) ψ ( α ) , ψ ( β ) (cid:1) for all α, β ∈ g .The theory of affine connections can readily be introduced, together with torsion-freeness and metric compatibility. Definition 3.8. Let ( A , g ) be a Lie-Rinehart algebra and let M be an A -module.A connection ∇ on M is a map ∇ : g → End K ( M ), written as α → ∇ α , such that(1) ∇ aα + β = a ∇ α + ∇ β (2) ∇ α ( am ) = a ∇ α m + α ( a ) m for all a ∈ A , α, β ∈ g and m ∈ M . Definition 3.9. Let ( A , g ) be a Lie-Rinehart algebra and let M be an A -modulewith connection ∇ and metric g . The connection is called metric if α (cid:0) g ( m , m ) (cid:1) = g ( ∇ α m , m ) + g ( m , ∇ α m )(3.4)for all α ∈ g and m , m ∈ M . Definition 3.10. Let ( A , g ) be a Lie-Rinehart algebra and let ∇ be a connectionon g . The connection is called torsion-free if ∇ α β − ∇ β α − [ α, β ] = 0for all α, β ∈ g .As in differential geometry, one can show that there exists a unique torsion-freeand metric connection associated to the Riemannian metric. The first step involvesproving Kozul’s formula. Proposition 3.11. Let ( A , g , g ) be a metric Lie-Rinehart algebra. If ∇ is a metricand torsion-free connection on g then it holds that g (cid:0) ∇ α β, γ (cid:1) = α (cid:0) g ( β, γ ) (cid:1) + β (cid:0) g ( γ, α ) (cid:1) − γ (cid:0) g ( α, β ) (cid:1) + g ( β, [ γ, α ]) + g ( γ, [ α, β ]) − g ( α, [ β, γ ])(3.5) for all α, β, γ ∈ g . JOAKIM ARNLIND AND AHMED AL-SHUJARY Proof. Starting from the right-hand-side of (3.5) and using the metric condition torewrite the first three terms as α (cid:0) g ( β, γ ) (cid:1) = g ( ∇ α β, γ ) + g ( β, ∇ α γ ) , together with the torsion-free condition to rewrite the last three terms as g ( β, [ γ, α ]) = g ( β, ∇ γ α ) − g ( β, ∇ α γ ) , immediately gives 2 g ( ∇ α β, γ ). (cid:3) By using Proposition 3.11 together with the fact that the metric is non-degenerate,one obtains the following result. Proposition 3.12. Let ( A , g , g ) be a metric Lie-Rinehart algebra. Then thereexists a unique metric and torsion-free connection on g .Remark . The unique connection in Proposition 3.12 will be referred to as the Levi-Civita connection of a metric Lie-Rinehart algebra. Proof. For every α, β ∈ g , the right-hand-side of (3.5) defines a linear form ω ∈ g ∗ ω ( γ ) = α (cid:0) g ( β, γ ) (cid:1) + β (cid:0) g ( γ, α ) (cid:1) − γ (cid:0) g ( α, β ) (cid:1) + g ( β, [ γ, α ]) + g ( γ, [ α, β ]) − g ( α, [ β, γ ]) . By assumption (see Definition 3.5), the metric induces an isomorphism map ˆ g : g → g ∗ , which implies that there exists an element ∇ α β = ˆ g − ( ω ) ∈ g suchthat g ( ∇ α β, γ ) = ω ( γ ). This shows that ∇ α β exists for all α, β ∈ g such thatrelation (3.5) is satisfied. Next, let us show that ∇ defines a connection on g , whichamounts to checking the four properties in Definition 3.8. This is a straight-forwardcomputation using (3.5) and the fact that, for instance, g ( ∇ aα β, γ ) = g ( a ∇ α β, γ ) for all γ ∈ g implies that ∇ aα β = a ∇ α β since the metric is non-degenerate. Let us illustratethe computation with the following example. From (3.5) it follows that2 g ( ∇ aα β, γ ) = aα (cid:0) g ( β, γ ) (cid:1) + β (cid:0) g ( γ, aα ) (cid:1) − γ (cid:0) g ( aα, β ) (cid:1) + g ( β, [ γ, aα ]) + g ( γ, [ aα, β ]) − g ( aα, [ β, γ ])= aα (cid:0) g ( β, γ ) (cid:1) + aβ (cid:0) g ( γ, α ) (cid:1) + β ( a ) g ( γ, α ) − aγ (cid:0) g ( α, β ) (cid:1) − γ ( a ) g ( α, β )+ g ( β, γ ( a ) α + a [ γ, α ]) + g ( γ, − β ( a ) α + a [ α, β ]) − ag ( α, [ β, γ ])= 2 ag ( ∇ α β, γ ) + β ( a ) g ( γ, α ) − γ ( a ) g ( α, β ) + γ ( a ) g ( β, α ) − β ( a ) g ( γ, α )= 2 ag ( ∇ α β, γ ) . The remaining properties of a connection is proved in an analogous way. To showthat ∇ is metric, one again uses (3.5) to substitute g ( ∇ α β, γ ) and g ( β, ∇ α γ ) andfind that α (cid:0) g ( β, γ ) (cid:1) − g ( ∇ α β, γ ) − g ( β, ∇ α γ ) = 0 . That the torsion-free condition holds follows from g ( ∇ α β, γ ) − g ( ∇ β α, γ ) − g ([ α, β ] , γ ) = 0 , which can be seen using (3.5). Hence, we conclude that there exists a metric andtorsion-free affine connection satisfying (3.5). Moreover, since the metric is non-degenerate, such a connection is unique. Finally, as every metric and torsion-free ¨AHLER–POISSON ALGEBRAS 7 connection on g satisfies (3.5) (by Proposition 3.11) we conclude that there existsa unique metric and torsion-free connection on g . (cid:3) In what follows, we shall recall some of the properties satisfied by a metric andtorsion-free connection. The differential geometric proofs goes through with onlya change in notation needed, but we provide them here for easy reference, andto adapt the formulation to our particular situation. We refer to [Koz60, Nel67]for a nice overview of differential geometric constructions in modules over generalcommutative algebras.Following the usual definitions, we introduce the curvature as R ( α, β ) γ = ∇ α ∇ β γ − ∇ β ∇ α γ − ∇ [ α,β ] γ (3.6)as well as R ( α, β, γ ) = R ( α, β ) γR ( α, β, γ, δ ) = g ( α, R ( γ, δ ) β ) . Let us also consider the extension of ∇ to multilinear maps T : g k → A ( ∇ β T )( α , . . . , α k ) = β (cid:0) T ( α , . . . , α k ) (cid:1) − k X i =1 T (cid:0) α , . . . , ∇ β α i , . . . , α k (cid:1) , as well as to g -valued multilinear maps T : g k → g ( ∇ β T )( α , . . . , α k ) = ∇ β (cid:0) T ( α , . . . , α k ) (cid:1) − k X i =1 T (cid:0) α , . . . , ∇ β α i , . . . , α k (cid:1) . As in classical geometry, one proceeds to derive the Bianchi identities. Proposition 3.14. Let ∇ be the Levi-Civita connection of a metric Lie-Rinehartalgbera ( A , g , g ) and let R denote corresponding curvature. Then it holds that R ( α, β, γ ) + R ( γ, α, β ) + R ( β, γ, α ) = 0 , (3.7) (cid:0) ∇ α R (cid:1) ( β, γ, δ ) + (cid:0) ∇ β R (cid:1) ( γ, α, δ ) + (cid:0) ∇ γ R (cid:1) ( α, β, δ ) = 0 , (3.8) for all α, β, γ, δ ∈ g .Proof. The first Bianchi identity (3.7) is proven by acting with ∇ γ on the torsionfree condition ∇ α β − ∇ β α − [ α, β ] = 0, and then summing over cyclic permutationsof α, β, γ . Since [[ α, β ] , γ ] + [[ β, γ ] , α ] + [[ γ, α ] , β ] = 0, the desired result follows.The second identity is obtained by a cyclic permutation (of α, β, γ ) in R (cid:0) ∇ α β −∇ β α − [ α, β ] , γ, δ (cid:1) = 0. One has0 = R (cid:0) ∇ α β − ∇ β α − [ α, β ] , γ, δ (cid:1) + cycl.= R ( ∇ γ α, β, δ ) + R ( α, ∇ γ β, δ ) − R ([ α, β ] , γ, δ ) + cycl.On the other hand, one has( ∇ γ R )( α, β, δ ) = ∇ γ R ( α, β, δ ) − R ( ∇ γ α, β, δ ) − R ( α, ∇ γ β, δ ) − R ( α, β, ∇ γ δ ) , and substituting this into the previous equation yields0 = ∇ γ R ( α, β, δ ) − (cid:0) ∇ γ R (cid:1) ( α, β, δ ) − R ( α, β, ∇ γ δ ) − R ([ α, β ] , γ, δ ) + cycl.After inserting the definition of R , and using that [[ α, β ] , γ ] + cycl. = 0, the secondBianchi identity follows. (cid:3) JOAKIM ARNLIND AND AHMED AL-SHUJARY Finally, one is able to derive the classical symmetries of the curvature tensor. Proposition 3.15. Let ∇ be the Levi-Civita connection of a metric Lie-Rinehartalgbera ( A , g , g ) and let R denote corresponding curvature. Then it holds that R ( α, β, γ, δ ) = − R ( β, α, γ, δ ) = − R ( α, β, δ, γ ) . (3.9) R ( α, β, γ, δ ) = R ( δ, γ, α, β ) , (3.10) for all α, β, γ, δ ∈ g .Proof. The identity R ( α, β, γ, δ ) = − R ( α, β, δ, γ ) follows immediately from the def-inition of R . Let us now prove that R ( α, β, γ, δ ) = − R ( β, α, γ, δ ). Starting from γ ( δ ( a )) − δ ( γ ( a )) − [ γ, δ ]( a ) = 0 and letting a = g ( α, β ) yields γ h g ( ∇ δ α, β ) + g ( α, ∇ δ β ) i − δ h g ( ∇ γ α, β ) + g ( α, ∇ γ β ) i − ( ∇ [ γ,δ ] α, β ) − ( α, ∇ [ γ,δ ] β ) = 0 . when using that ∇ is a metric connection; i.e τ ( g ( α, β )) = g ( ∇ τ α, β ) + g ( α, ∇ τ β )for τ = γ, δ, [ γ, δ ]. A further expansion using the metric property gives g ( ∇ γ ∇ δ α, β ) + g ( α, ∇ γ ∇ δ β ) − g ( ∇ δ ∇ γ α, β ) − g ( α, ∇ δ ∇ γ β ) − g ( ∇ [ γ,δ ] α, β ) − g ( α, ∇ [ γ,δ ] β ) = 0 , which is equivalent to g ( R ( γ, δ ) α, β ) = − g ( R ( γ, δ ) β, α ) . Next, one can make use of equation (3.7) in Proposition 3.14, from which it followsthat R ( α, β, γ, δ ) + R ( α, δ, β, γ ) + R ( α, γ, δ, β ) = 0 . (3.11)It is a standard algebraic result that any quadri-linear map satisfying (3.9) and(3.11) also satisfies (3.10) (see e.g. [Hel01]). (cid:3) K¨ahler–Poisson algebras In this section, we shall introduce a type of Poisson algebras, that resembles thesmooth functions on an (isometrically) embedded almost K¨ahler manifold, in sucha way that an analogue of Riemannian geometry may be developed. Namely, letus consider a unital Poisson algebra ( A , {· , ·} ) and let { x , . . . , x m } be a set ofdistinguished elements of A , corresponding to functions providing an embeddinginto R m , in the geometrical case. One may also consider the setting of algebraic(Poisson) varieties where A is a finitely generated Poisson algebra and { x , . . . , x m } denotes a set of generators. Our aim is to introduce equation (2.2) in A andinvestigate just how far one may take the analogy with Riemannian geometry. Afterintroducing K¨ahler–Poisson algebras below, we will show that they are, in a naturalway, metric Lie-Rinehart algebras, which implies that the results of Section 3 canbe applied; in particular, there exists a unique torsion-free metric connection onevery K¨ahler–Poisson algebra. Note that Lie-Rinehart algebras related to Poissonalgebras have extensively been studied by Huebschmann (see e.g. [Hue90, Hue99]).In Section 2 it was shown that the following identity holds on an almost K¨ahlermanifold: γ { f, h } = −{ f, x i } ¯ g ij { x j , x k } ¯ g kl { x l , h } . (2.2) ¨AHLER–POISSON ALGEBRAS 9 This equation is well-defined in a Poisson algebra, and we shall use it to define themain object of our investigation. Definition 4.1. Let A be a Poisson algebra over K and let { x , . . . , x m } ⊆ A .Given g ij ∈ A , for i, j = 1 , . . . , m , such that g ij = g ji , we say that the triple K = (cid:0) A , { x , . . . , x m } , g (cid:1) is a K¨ahler–Poisson-algebra if there exists η ∈ A suchthat m X i,j,k,l =1 η { a, x i } g ij { x j , x k } g kl { x l , b } = −{ a, b } (4.1)for all a, b ∈ A . Remark . From now on, we shall use the differential geometric convention thatrepeated indices are summed over from 1 to m , and omit explicit summation sym-bols.Given a K¨ahler–Poisson-algebra K , we let g denote the A -module generated by allinner derivations, i.e. g = { a { c , ·} + · · · + a N { c N , ·} : a i , c i ∈ A and N ∈ N } . It is a standard fact that g is a Lie algebra over K with respect to[ α, β ]( a ) = α (cid:0) β ( a ) (cid:1) − β (cid:0) α ( a ) (cid:1) . The matrix g induces a bilinear symmetric form on g , defined by g ( α, β ) = α ( x i ) g ij β ( x j ) , (4.2)and we refer to g as the metric on g . To the metric g one may associate the mapˆ g : g → g ∗ defined as ˆ g ( α )( β ) = g ( α, β ) . Proposition 4.3. If K = (cid:0) A , { x , . . . , x m } , g (cid:1) is a K¨ahler–Poisson-algebra thenthe metric g is non-degenerate; i.e. the map ˆ g : g → g ∗ is a module isomorphism.Proof. Let us first show that g is injective; i.e. we will show that ˆ g ( α )( β ) = 0,for all β ∈ g , implies that α = 0. Thus, write α = α i { x i , ·} , and assume that g ( α, β ) = 0 for all β ∈ g . In particular, we can choose β = η { c, x k } g km {· , x m } , forarbitrary c ∈ A , which implies that0 = g ( α, β ) = ηα k { x k , x i } g ij { c, x k } g km { x j , x m } = − α k η { x k , x i } g ij { x j , x m } g mk { x k , c } . Using the relation (4.1), one obtains α k { x k , c } = 0for all c ∈ A , which is equivalent to α = 0. This shows that ˆ g is injective. Let usnow show that ˆ g is surjective. Thus, let ω ∈ g ∗ and set α = ηω ( { x i , ·} ) g ij { x j , ·} ∈ g , which gives ˆ g ( α )( a k { b k , ·} ) = ηω ( { x i , ·} ) g ij { x j , x l } g lm a k { b k , x m } = − ηa k { b k , x m } g ml { x l , x j } g ji ω ( { x i , ·} ) . Since ω is a module homomorphism one obtainsˆ g ( α )( a k { b k , ·} ) = ω ( − ηa k { b k , x m } g ml { x l , x j } g ji { x i , ·} )= ω ( a k { b k , ·} ) , by using (4.1), which proves that every element of g ∗ is in the image of ˆ g . Weconclude that ˆ g is a module isomorphism. (cid:3) Corollary 4.4. If ( A , { x , . . . , x m } , g ) is a K¨ahler–Poisson algebra then ( A , g , g ) is a metric Lie-Rinehart algebra.Proof. It is easy to check that ( A , g ) satisfies the conditions of a Lie-Rinehartalgebra, and Proposition 4.3 implies that the metric is non-degenerate. Hence,( A , g , g ) is a metric Lie-Rinehart algebra. (cid:3) Let us now introduce some notation for K¨ahler–Poisson algebras. Thus, we set P ij = { x i , x j }P i ( a ) = { x i , a } , for a ∈ A , as well as D ij = η P ik P jk = η { x i , x l } g lk { x j , x k }D i ( a ) = η P k ( a ) P ki = η { x k , a } g kl { x l , x i } , and note that D ij = D ji . With respect to this notation, (4.1) can be stated as D i ( a ) P i ( b ) = { a, b } . (4.3)The metric will be used to lower indices in analogy with differential geometry. E.g. P ij = P ik g kj D ij = D ik g kj . Furthermore, one immediately derives the following useful identities D ij P j ( a ) = P i ( a ) , P ij D j ( a ) = P i ( a ) and D ij D jk = D ik . (4.4)by using (4.1).There is a natural embedding ι : g → A m , given by ι ( a i { b i , ·} ) = a i { b i , x k } e k , where { e k } mk =1 denotes the canonical basis of the free module A m . Moreover, g defines a bilinear form on A m via g ( X, Y ) = X i g ij Y j for X = X i e i ∈ A m and Y = Y i e i ∈ A m , and we introduce the map D : A m → A m by setting D ( X ) = D ij X j e i for X = X i e i ∈ A m . Proposition 4.5. The map D : A m → A m is an orthogonal projection; i.e. D ( X ) = D ( X ) and g ( D ( X ) , Y ) = g ( X, D ( Y )) for all X, Y ∈ A m . ¨AHLER–POISSON ALGEBRAS 11 Proof. First, it is clear that D is an endomorphism of A m . It follows immediatelyfrom (4.4) that D ( X ) = D ij D jk X k e i = D ij D jl g lk X k e i = D il g lk X k e i = D ik X k e i = D ( X ) . Furthermore, using that D ij = D ji one finds that g (cid:0) D ( X ) , Y (cid:1) = D ij X j g ik Y k = X j D il g lj g ik Y k = X j g lj D li g ik Y k = X j g jl D lk Y k = g (cid:0) X, D ( Y ) (cid:1) , which completes the proof. (cid:3) From Proposition 4.5 we conclude that T A = im( D )is a finitely generated projective module. As a corollary, we prove that g is a finitelygenerated projective module by showing that g is isomorphic to T A . Proposition 4.6. The map ι : g → A m is an isomorphism from g to T A .Proof. First, it is clear from the definition that ι is a module homomorphism.Considered as a submodule of A m , elements of T A can be characterized by the factthat D ( X ) = X for all X ∈ T A . Thus, by showing that D (cid:0) ι ( a k { b k , ·} ) (cid:1) = D ij a k { b k , x j } e i = − a k D ij P j ( b k ) a k e i = − a k P i ( b k ) = ι ( a k { b k , ·} )it follows that ι ( a k { b k , ·} ) ∈ T A . Let us now show that ι is injective; assume that ι ( a k { b k , ·} ) = 0, which implies that a k { b k , x i } = 0 for i = 1 , . . . , m. Next, for arbitrary c ∈ A , we write a k { b k , c } = − ηa k { b k , x i } g ij P jl g lm { x m , c } , by using (4.1). Since a k { b k , x i } = 0, one obtains a k { b k , c } = 0 for all c ∈ A .To prove that ι is surjective, we start from an arbitrary X = X i e i ∈ T A , andnote that ι (cid:0) X i g ij D i ( · ) (cid:1) = X i g ij D ik e k = D ( X ) = X by using that D ( X ) = X for all X ∈ T A . Hence, we may conclude that ι is anisomorphism from g to T A . (cid:3) Corollary 4.7. g is a finitely generated projective module. Note that the above result is clearly not dependent on whether or not the underly-ing Poisson algebra has the structure of a K¨ahler–Poisson algebra, as the definitionof g involves only inner derivations. Hence, as soon as the Poisson algebra ad-mits the structure of a K¨ahler–Poisson algebra, it follows that the module of innerderivations is projective. Furthermore, the fact that g is a projective module hasseveral implications for the underlying Lie-Rinehart algebra [Rin63, Hue90]. Next,let us show that the derivations D i generate g as an A -module. Proposition 4.8. The A -module g is generated by {D , . . . , D m } . Proof. First of all, it is clear that every element in the module generated by D i ,written as α ( c ) = α i D i ( c ) = ηα i { x i , x j } g jk { c, x k } , is an element of g . Conversely, let α ∈ g be an arbitrary element written as α ( c ) = X N a N { b N , c } . for c ∈ A . Using the K¨ahler–Poisson condition (4.1) one may write this as α ( a ) = X N a N { b N , c } = − X N ηa N { b N , x i } g ij { x j , x k } g kl { x l , c } = (cid:16) X N a N { b N , x i } g ij (cid:17) D j ( c ) , which clearly lies in the module generated by {D , . . . , D m } . (cid:3) Thus, every α ∈ g may be written as α = α i D i for some α i ∈ A . It turns out thatthis is a very convenient way of writing elements of g , which shall be extensivelyused in the following. Note that if the K¨ahler–Poisson algebra comes from an almostK¨ahler manifold M , then D i is quite close to a partial derivative on M in the sensethat ( ∂ a x i ) g ik D k ( f ) = ∂ a f , for f ∈ C ∞ ( M ).4.1. The trace of linear maps. As we shall be interested in both Ricci and scalarcurvature, which are defined using traces of linear maps, we introducetr( L ) = g (cid:0) L ( D i ) , D j (cid:1) D ij . (4.5)for an A -linear map L : g → g . This trace coincides with the ordinary trace on g ∗ ⊗ A g ; namely, consider L = X N ω N ⊗ A α N ∈ g ∗ ⊗ A g as a linear map L : g → g in the standard way via L ( β ) = X N ω N ( β ) α N , together with tr( L ) = X N ω N ( α N ) . Writing α N = α Ni D i one finds that g (cid:0) L ( D i ) , D j (cid:1) D ij = X N g (cid:0) ω N ( D i ) α Nk D k , D j (cid:1) D ij = X N ω N ( D i ) α Nk D kj D ij = X N ω N ( α Nk D ki D i ) = X N ω N ( α Nk D k ) = X N ω N ( α N ) . In particular, this implies that the trace defined via (4.5) is independent of theK¨ahler–Poisson structure. ¨AHLER–POISSON ALGEBRAS 13 Morphisms of K¨ahler–Poisson algebras. As K¨ahler–Poisson algebras arealso metric Lie-Rinehart algebras, we shall require that a morphism of K¨ahler–Poisson algebras is also a morphism of metric Lie-Rinehart algebras (as definedin Section 3). However, as the definition of a K¨ahler–Poisson also involves thechoice of a set of distinguished elements, we will require a morphism to respectthe subalgebra generated by these elements. To this end, we start by making thefollowing definition. Definition 4.9. Given a K¨ahler–Poisson algebra ( A , { x , . . . , x m } , g ), let A fin ⊆ A denote the subalgebra generated by { x , . . . , x m } .Equipped with this definition, we introduce morphisms of K¨ahler–Poisson algebrasin the following way. Definition 4.10. Let K = ( A , { x , . . . , x m } , g ) and K ′ = ( A ′ , { y , . . . , y m ′ } , g ′ ) beK¨ahler–Poisson algebras together with their corresponding modules of derivations g and g ′ , respectively. A morphism of K¨ahler–Poisson algebras is a pair of maps( φ, ψ ), with φ : A → A ′ and ψ : g → g ′ , such that ( φ, ψ ) is a morphism of themetric Lie-Rinehart algebras ( A , g , g ) and ( A , g ′ , g ′ ) and φ is a Poisson algebrahomomorphism such that φ ( A fin ) ⊆ A ′ fin .Note that if the algebras are finitely generated such that A = A fin and A ′ = A ′ fin (which is the case in many examples), the condition φ ( A fin ) ⊆ A ′ fin is automaticallysatisfied. Although a morphism of K¨ahler–Poisson algebras is given by a choice oftwo maps φ and ψ , it is often the case that φ determines ψ in the following sense. Proposition 4.11. Let ( φ, ψ ) : ( A , { x , . . . , x m } , g ) → ( A ′ , { y , . . . , y m ′ } , g ′ ) be amorphism of K¨ahler–Poisson algebras such that for all α ′ ∈ g ′ α ′ (cid:0) φ ( a ) (cid:1) = 0 ∀ a ∈ A ⇒ α ′ = 0 then ψ (cid:0) a { b, ·} A (cid:1) = φ ( a ) { φ ( b ) , ·} A ′ . Proof. Let ( φ, ψ ) : ( A , { x , . . . , x m } , g ) → ( A ′ , { y , . . . , y m ′ } , g ′ )be a morphism of K¨ahler–Poisson algebras fulfilling the assumption above. Since φ is a Poisson algebra homomorphism, one obtains for α = a { b, ·} A φ (cid:0) α ( c ) (cid:1) = φ (cid:0) a { b, c } A (cid:1) = φ ( a ) { φ ( b ) , φ ( c ) } A ′ for all a ∈ A . By the definition of a Lie-Rinehart morphism, this has to equal ψ ( α )( φ ( c )); i.e. ψ ( α )( φ ( c )) = φ ( a ) { φ ( b ) , φ ( c ) } A ′ . Thus, ψ ( α ) agrees with φ ( a ) { φ ( b ) , ·} A ′ on the image of φ , which implies that ψ ( α ) = φ ( a ) { φ ( b ) , ·} A ′ since any derivation is determined by its action on the image of φ by assumption. (cid:3) For instance, the requirements in Proposition 4.11 are clearly satisfied if φ is sur-jective. Construction of K¨ahler–Poisson algebras. Given a Poisson algebra ( A , {· , ·} )one may ask if there exist { x , . . . , x m } and g ij such that ( A , { x , . . . , x m } , g ) is aK¨ahler–Poisson algebra? Let us consider the case when A is a finitely generatedalgebra, and let { x , . . . , x m } be an arbitrary set of generators. If we denote by P the matrix with entries { x i , x j } and by g the matrix with entries g ij , the K¨ahler–Poisson condition (4.1) may be written in matrix notation as η P g P g P = −P . Given an arbitrary antisymmetric matrix P , we shall find g by first writing P ina block diagonal form, with antisymmetric 2 × Lemma 4.12. Let M N ( R ) denote the set of N × N matrices with entries in R . For N ≥ , let P ∈ M N ( R ) be an antisymmetric matric. Then there exists V ∈ M N ( R ) ,an antisymmetric Q ∈ M N − ( R ) and λ ∈ R such that V T P V = λ − λ Q .Proof. We shall construct the matrix V by using elementary row and column op-erations. Note that if a matrix E represents an elementary row operation, then E T P E is obtained by applying the elementary operation to both the row and thecorresponding column. Denoting the matrix elements of P by p ij , we start by con-structing a matrix V k such that ( V Tk P V k ) k = ( V Tk P V k ) k = 0 (which necessarilyimplies that also the (1 k ) and (2 k ) matrix elements are zero). To this end, let V k denote the matrix representing the elementary row operation that multiplies the k ’th row by p , and let V k represent the operation that adds the first row, multi-plied by − p k , to the k ’th row. Furthermore, V k represents the operation of addingthe second row, multiplied by p k , to the k ’th row. Setting V k = V k V k V k it is easyto see that V Tk P V k is an antisymmetric matrix where the (1 k ), (2 k ), ( k 1) and ( k V = V V · · · V N and conclude that V T P V is of the desired form. (cid:3) Proposition 4.13. Let P ∈ M N ( R ) be an antisymmetric matric, and let ˆ N denotethe integer part of N/ . Then there exists V ∈ M N ( R ) and λ , . . . , λ ˆ N ∈ R suchthat V T P V = diag(Λ , . . . , Λ ˆ N ) if N is even, V T P V = diag(Λ , . . . , Λ ˆ N , if N is odd,where Λ k = (cid:18) λ k − λ k (cid:19) . Proof. Let us prove the statement by using induction together with Lemma 4.12.Thus, assume that there exists V ∈ M N ( R ) such that V T P V = diag(Λ , . . . , Λ k , Q k +1 )where Q k +1 ∈ M N − k is an antisymmetric matrix. Clearly, by Lemma 4.12, thisholds true for k = 1. Next, assume that N − k ≥ 2. Applying Lemma 4.12 to ¨AHLER–POISSON ALGEBRAS 15 Q k +1 we conclude that there exists V k +1 ∈ M N − k ( R ) such that V Tk +1 Q k +1 V k +1 =diag(Λ k +1 , Q k +2 ). Furthermore, defining W k +1 ∈ M N ( R ) by W k +1 = diag( k , V k +1 )one finds that ( V W k +1 ) T P ( V W k +1 ) = diag(Λ , . . . , Λ k +1 , Q k +2 ) . By induction, it follows that one may repeat this procedure until N − k < 2. If N is even, then N − k = 0 and the statement follows. If N is odd, then N − k = 1and, since V T P V is antisymmetric, it follows that the ( N N ) matrix element iszero, giving the stated result. (cid:3) Returning to the case of a Poisson algebra generated by x , . . . , x m , assume for themoment that m = 2 N for a positive integer N . By Proposition 4.13, there exists amatrix V V T P V = P where P is a block diagonal matrix of the form P = diag(Λ , . . . , Λ N )with Λ k = (cid:18) λ k − λ k (cid:19) . In the same way, defining g = diag( g , . . . , g N ) with g k = λλ k (cid:18) (cid:19) λ = λ · · · λ N we set g = V g V T . Noting that P g P g P = − λ P one finds0 = P g P g P + λ P = V T P V g V T P V gV T P V + λ V T P V = V T (cid:0) P g P g P + λ P (cid:1) V It is a general fact that for an arbitrary matrix V there exists a matrix ˜ V such that˜ V V = V ˜ V = (det V ) . Multiplying the above equation from the left by ˜ V T andfrom the right by ˜ V yieldsdet( V ) (cid:0) P g P g P + λ P (cid:1) = 0 . (4.6)As long as det( V ) is not a zero divisor, this implies that P g P g P = − λ P . Thus, given a finitely generated Poisson algebra A , the above procedure gives arather general way to associate a localization A [ λ − ] and a metric g to A , suchthat ( A [ λ − ] , { x , . . . , x m } , g ) is a K¨ahler–Poisson algebra. Note that the aboveargument, with only slight notational changes, also applies to the case when m isodd, in which case an extra block of 0 will appear in P . The Levi-Civita connection Since every K¨ahler–Poisson algebra is also a metric Lie-Rinehart algebra, the resultsof Section 3 immediately applies. In particular, there exists a unique torsion-freeand metric connection on the module g . In this section, we shall derive an explicitexpression for the Levi-Civita connection of an arbitrary K¨ahler–Poisson algebra.It turns out to be convenient to formulate the results in terms of the generators {D , . . . , D m } . Kozul’s formula gives the connection as2 g ( ∇ D i D j , D k ) = D i (cid:0) g ( D j , D k ) (cid:1) + D j (cid:0) g ( D k , D i ) (cid:1) − D k (cid:0) g ( D i , D j ) (cid:1) − g ([ D j , D k ] , D i ) + g ([ D k , D i ] , D j ) + g ([ D i , D j ] , D k ) , (5.1)and one notes that an element α = a { b, ·} ∈ g may be recovered from g ( α, D i ) as g ( α, D i ) D i ( f ) = a { b, x k }D ik D i ( f ) = a { b, x k }D k ( f ) = a { b, f } = α ( f ) . Thus, one immediately obtains ∇ D i D j = g ( ∇ D i D j , D k ) D k . However, it turns outthat one can obtain a more compact formula for the connection. Let us start byproving the following result. Lemma 5.1. g ([ D i , D j ] , D k ) = D i (cid:0) D jk (cid:1) − D j (cid:0) D ik (cid:1) .Proof. For convenience, let us introduce the notation ˆ P ij = η { x i , x j } and, conse-quently, ˆ P ij = ˆ P ik g kj . In this notation, one finds D i ( a ) = ˆ P ij { a, x j } . Thus, oneobtains g ([ D i , D j ] , D k ) = [ D i , D j ]( x l ) D kl = ˆ P im {D jl , x m }D kl − ˆ P jn {D il , x n }D kl = (cid:0) ˆ P im { ˆ P jn { x l , x n } , x m } − ˆ P jn { ˆ P im { x l , x m } , x n } (cid:1) D kl = ˆ P im ˆ P jn (cid:0) − {{ x n , x l } , x m } − {{ x l , x m } , x n } (cid:1) D kl + (cid:0) ˆ P im { ˆ P jn , x m }{ x l , x n } − ˆ P jn { ˆ P im , x n }{ x l , x m } (cid:1) D kl = ˆ P im ˆ P jn {{ x m , x n } , x k } + ˆ P im { ˆ P jn , x m }{ x k , x n }− ˆ P jn { ˆ P im , x n }{ x k , x m } , by using the Jacobi identity together with { a, x i }D ki = { a, x k } . Furthermore, inthe second and third term, one uses Leibniz’s rule to obtain g ([ D i , D j ] , D k ) = ˆ P im ˆ P jn {{ x m , x n } , x k } + ˆ P im { ˆ P jn { x k , x n } , x m }− ˆ P im ˆ P jn {{ x k , x n } , x m } − ˆ P jn { ˆ P im { x k , x m } , x n } + ˆ P jn ˆ P im {{ x k , x m } , x n } = ˆ P im ˆ P jn (cid:0) {{ x m , x n } , x k } + {{ x n , x k } , x m } + {{ x k , x m } , x n } (cid:1) + D i (cid:0) D jk (cid:1) − D j ( D ik ) = D i (cid:0) D jk (cid:1) − D j ( D ik ) , by again using the Jacobi identity. (cid:3) The above result allows for the following formulation of the Levi-Civita connectionfor a K¨ahler–Poisson algebra. Proposition 5.2. If ∇ denotes the Levi-Civita connection of a K¨ahler–Poissonalgebra K then ∇ D i D j = 12 D i ( D jk ) D k − D j ( D ik ) D k + 12 D k ( D ij ) D k , (5.2) ¨AHLER–POISSON ALGEBRAS 17 or, equivalently, ∇ D i D j = Γ ij k D k where Γ ij k = 12 D i ( D jl ) D lk − D j ( D il ) D lk + 12 D k ( D ij ) . (5.3) Proof. Since g ( D i , D j ) = D ij , Kozul’s formula (5.1) together with Lemma 5.1 gives2 g ( ∇ D i D j , D k ) = D i ( D jk ) + D j ( D ki ) − D k ( D ij ) − D j ( D ki ) + D k ( D ji )+ D k ( D ij ) − D i ( D kj ) + D i ( D jk ) − D j ( D ik )= D i ( D jk ) − D j ( D ki ) + D k ( D ij ) , which proves (5.2). The fact that one may write the connection as ∇ D i D j = Γ ijk D k follows from D ij D j = D i and D k ( a ) D k ( b ) = D k ( a ) D k ( b ). (cid:3) Thus, for arbitrary elements of g , one obtains ∇ α β = α ( β i ) D i + Γ ijk α i β j D k (5.4)where α = α i D i and β = β i D i , and curvature is readily introduced as R ( α, β ) γ = ∇ α ∇ β γ − ∇ β ∇ α γ − ∇ [ α,β ] γ. Ricci curvature is defined asRic( α, β ) = tr (cid:0) γ → R ( γ, α ) β (cid:1) and using the trace from Section 4.1, one obtainsRic( α, β ) = g ( R ( D i , α ) β, D j ) D ij . To define the scalar curvature, one considers the Ricci curvature as a linear mapRic : g → g with Ric( α ) = Ric( α, D i ) D i , giving S = tr (cid:0) α → Ric( α ) (cid:1) = g (cid:0) R ( D i , D k ) D l , D j (cid:1) D ij D kl . Note that since the metric is nondegenerate, there exists a unique element ∇ f ∈ g such that g ( ∇ f, α ) = α ( f ) for all α ∈ g ; we call ∇ f the gradient of f . Now, it iseasy to see that ∇ f = D i ( f ) D i since g ( D i ( f ) D i , α j D j ) = D i ( f ) α j D ij = α j D j ( f ) = α ( f ) . The divergence of an element α ∈ g is defined asdiv( α ) = tr( β → ∇ β α ) , and, finally, the Laplacian ∆( f ) = div( ∇ f ) . Examples As shown in Section 2, the algebra of smooth functions on an almost K¨ahler man-ifold M becomes a K¨ahler–Poisson algebra when choosing x , . . . , x m to be em-bedding coordinates, providing an isometric embedding into R m , endowed withthe standard Euclidean metric. (Recall that, by Nash’s theorem [Nas56], such anembedding always exists.) In this section, we shall present examples of a morealgebraic nature to illustrate the fact that algebras of smooth functions are not theonly examples of K¨ahler–Poisson algebras.Keeping in mind the general construction procedure in Section 4.3, we considerfinitely generated Poisson algebras with a low number of generators.6.1. Poisson algebras generated by two elements. Let A be a unital Poissonalgebra generated by the two elements x = x ∈ A and x = y ∈ A , and set P = (cid:18) { x, y }−{ x, y } (cid:19) It is easy to check that for an arbitrary symmetric matrix g P g P g P = −{ x, y } det( g ) P . Thus, as long as { x, y } det( g ) is not a zero-divisor, one may localize to obtain aK¨ahler–Poisson algebra K = ( A [( { x, y } det( g )) − ] , { x, y } , g ) . For the sake of illustrating the concepts and formulas we have developed so far,let us explicitly work out an example based on an algebra A , generated by twoelements. Let us start by choosing an element λ ∈ A for which the localization A = A [ p − , λ − ] exists, and then defining the metric as g = 1 λ (cid:18) (cid:19) From the above considerations, we know that ( A , { x, y } , g ) is a K¨ahler–Poissonalgebra with η = λ /p , where p = { x, y } . For convenience we also introduce γ = p/λ such that η = 1 /γ . Let us start by computing the derivations D x = D and D y = D , which generate the module g : D x = η { x, x i } g ij {· , x j } = λp {· , y } = − γ { y, ·}D y = η { y, x i } g ij {· , x j } = − λp {· , x } = 1 γ { x, ·} as well as D x = g k D k = 1 λ D x and D y = g k D k = 1 λ D y . Moreover, they provide an orthogonal set of generators since g ( D x , D x ) = 1 γ { y, x i } g ij γ { y, x j } = 1 γ p λ = λg ( D y , D y ) = λ g ( D x , D y ) = 0 , ¨AHLER–POISSON ALGEBRAS 19 and one obtains ( D ij ) = (cid:0) g ( D i , D j ) (cid:1) = (cid:18) λ λ (cid:19) . Note that g is a free module with basis {D x , D y } since a D x + b D y = 0 ⇒ ( a D x ( x ) + b D y ( x ) = 0 a D x ( y ) + b D y ( y ) = 0 ⇒ ( − a γ { y, x } = 0 b γ { x, y } = 0 ⇒ ( a = 0 b = 0by using that λ is invertible.Let us introduce the derivation D λ = γ − { λ, ·} and note that D λ = [ D x , D y ] = 1 λ D x ( λ ) D y − λ D y ( λ ) D x . From Proposition 5.2 one computes the connection: ∇ D x D x = 12 D ( D k ) D k − D ( D k ) D k + 12 D k ( D ) D k = 12 D x ( λ ) D x + 12 D y ( λ ) D y = 12 D i ( λ ) D i and similarly ∇ D y D y = 12 D x ( λ ) D x + 12 D y ( λ ) D y = ∇ D x D x ∇ D x D y = 12 D x ( λ ) D y − D y ( λ ) D x = D λ ∇ D y D x = 12 D y ( λ ) D x − D x ( λ ) D y = −D λ Moreover, the curvature can readily be computed R ( D x , D y ) D x = (cid:20) D x ( λ ) + D y ( λ ) − D x (cid:0) D x ( λ ) (cid:1) − D y (cid:0) D y ( λ ) (cid:1)(cid:21) D y R ( D x , D y ) D y = − (cid:20) D x ( λ ) + D y ( λ ) − D x (cid:0) D x ( λ ) (cid:1) − D y (cid:0) D y ( λ ) (cid:1)(cid:21) D x , as well as the scalar curvature S = 1 λ (cid:16) D x (cid:0) D x ( λ ) (cid:1) + D y (cid:0) D y ( λ ) (cid:1) − D x ( λ ) − D y ( λ ) (cid:17) . Moreover, one finds that ∇ f = D x ( f ) D x + D y ( f ) D y div( α x D x + α y D y ) = D x ( α x ) + D y ( α y )∆( f ) = D x (cid:0) D x ( f ) (cid:1) + D y (cid:0) D y ( f ) (cid:1) = D x (cid:0) D x ( f ) (cid:1) + D y (cid:0) D y ( f ) (cid:1) − D x ( λ ) D x ( f ) − D y ( λ ) D y ( f ) . Poisson algebras generated by three elements. Let A be a unital Poissonalgebra generated by x = x, x = y, x = z ∈ A . Writing { x, y } = a , { y, z } = b and { z, x } = c , i.e. P = a − c − a bc − b , one readily checks that for an arbitrary symmetric matrix g P g P g P = − τ P with τ = a | g | + b | g | + c | g | + 2 ab | g | − ac | g | − bc | g | , where | g | ij denotes the determinant of the matrix obtained from g by deleting the i ’th row and the j ’th column. Thus, one may construct the K¨ahler–Poisson algebra K = {A [ τ − ] , { x, y, z } , g } . In particular, if g = diag( λ, λ, λ ), then τ = λ ( a + b + c ).Let us now construct a particular class of algebras with a natural geometricinterpretation and a close connection to algebraic geometry. Let R [ x, y, z ] be thepolynomial ring in three variables over the real numbers, and write x = x , x = y and x = z . For arbitrary C ∈ R [ x, y, z ], it is straight-forward to show that { x i , x j } = ε ijk ∂ k C, where ε ijk denotes the totally antisymmetric symbol with ε = 1, defines a Poissonstructure on R [ x, y, z ] which is well-defined on the quotient A C = R [ x, y, z ] / ( C )since { x i , C } = { x i , x j } ∂ j C = ε ijk ( ∂ k C )( ∂ j C ) = 0 . In the spirit of algebraic geometry, the algebra A C has a natural interpretationas the polynomial functions on the level set C ( x, y, z ) = 0 in R . Choosing themetric g ij = δ ij (corresponding to the Euclidean metric on R ) one obtains aK¨ahler–Poisson algebra ( b A C , { x, y, z } , g ) where b A C = A C [ τ − ] and τ = (cid:0) ∂ x C (cid:1) + (cid:0) ∂ y C (cid:1) + (cid:0) ∂ z C (cid:1) , with η = τ − . Note that the points in R , for which τ ( x, y, z ) = 0, coincide withthe singular points of C ( x, y, z ) = 0; i.e. points where ∂ x C = ∂ y C = ∂ z C = 0.As an illustration, let us choose C = ( ax + by + cz − ) for a, b, c ∈ R , giving { x, y } = cz, { y, z } = ax and { z, x } = by. and η = (cid:0) a x + b y + c z (cid:1) − together with ( D ij ) = η b y + c z − abxy − acxz − abxy a x + c z − bcyz − acxz − bcyz a x + b y . A straight-forward, but somewhat lengthy, calculation gives R ( D x , D y ) D x D y D z = cz b R D x D y D z R ( D y , D z ) D x D y D z = ax b R D x D y D z R ( D z , D x ) D x D y D z = by b R D x D y D z where b R = abcη − cz bycz − ax − by ax , ¨AHLER–POISSON ALGEBRAS 21 and the scalar curvature becomes S = 2 abcη . Summary In this note, we have introduced the concept of K¨ahler–Poisson algebras as a meanto study Poisson algebras from a metric point of view. As shown, the single relation(4.1) has consequences that allow for an identification of geometric objects in thealgebra, which share crucial properties with their classical counterparts. The ideabehind the construction was to identify a distinguished set of elements in the algebrathat serve as “embedding coordinates”, and then construct the projection operator D that projects from the tangent space of the ambient manifold onto that of theembedded submanifold. It is somewhat surprising that (4.1) encodes the crucialelements that are needed for the algebra to resemble an algebra of functions on analmost K¨ahler manifold.As outlined in Section 4.3, a large class of Poisson algebras admit a K¨ahler–Poisson algebra as an associated localization, which shows a certain generality ofour treatment. Thus, even if one is not interested in metric structures on a Poissonalgebra, the tools we have developed might be of help. For instance, if a Poissonalgebra can be given the structure of a K¨ahler–Poisson algebra, one immediatelyconcludes that the module generated by the inner derivations is a finitely generatedprojective module. A statement which is clearly independent of any metric struc-ture. A comparison with differential geometry is close at hand, where the structureof a Riemannian manifold can be used to prove results about the underlying man-ifold (or even the topological structure).Let us end with a brief outlook. After having studied the basic properties ofK¨ahler–Poisson algebras in this paper, there are several natural questions thatcan be studied. For instance, what is the interplay between the cohomology (ofLie-Rinehart algebras) and the Levi-Civita connection? Can one perhaps use theconnection to compute cohomology? Is there a natural way to study the modulispaces of Poisson algebras; i.e. how many (non-isomorphic) K¨ahler–Poisson struc-tures does there exist on a given Poisson algebra? We hope to return to these, andmany other interesting questions, in the near future. Acknowledgments We would like to thank M. Izquierdo for ideas and discussions. Furthermore, J. A.is supported by the Swedish Research Council. References [AH14] J. Arnlind and G. Huisken. Pseudo-Riemannian geometry in terms of multi-linear brack-ets. Lett. Math. Phys. , 104(12):1507–1521, 2014.[AHH12] J. Arnlind, J. Hoppe, and G. Huisken. Multi-linear formulation of differential geometryand matrix regularizations. J. 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Differential forms on general commutative algebras. Trans. Amer. Math.Soc. , 108:195–222, 1963.[Wei83] A. Weinstein. The local structure of Poisson manifolds. J. Differential Geom. , 18(3):523–557, 1983.(Joakim Arnlind) Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden E-mail address : [email protected] (Ahmed Al-Shujary) Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden E-mail address ::