arXiv:1712.08397v1 [math.RA] 22 Dec 2017

KÄHLER–POISSON ALGEBRAS
JOAKIM ARNLIND AND AHMED AL-SHUJARY
Abstract. We introduce Kähler–Poisson algebras as analogues of algebras of
smooth functions on Kähler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the
module of inner derivations of a Kähler–Poisson algebra is a finitely generated
projective module, and allows for a unique metric and torsion-free connection
whose curvature enjoys all the classical symmetries. Moreover, starting from
a large class of Poisson algebras, we show that every algebra has an associated
Kähler–Poisson algebra constructed as a localization. At the end, detailed
examples are provided in order to illustrate the novel concepts.

1. Introduction
Poisson manifolds and their geometry have been of great interest over the last
decades. Besides from being important from a purely mathematical point of view,
they are also fundamental to areas in mathematical and theoretical physics. Many
authors have studied the geometric and algebraic properties of symplectic and Poisson manifolds together in relation to concepts such as connections, local structure
and cohomology (see e.g. [Lic77, Wei83, Bry88, Hue90]). Moreover, there is a well
developed field of deformations of Poisson structures, perhaps most famous through
Kontsevich’s result on the existence of formal deformations [Kon03]. The ring of
smooth functions on a Poisson manifold is a Poisson algebra and it seems quite natural to ask to what extent geometric properties and concepts may be introduced in
an 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 shall
start 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 geometry of an embedded Kähler manifold M entirely in terms of the Poisson structure
on the algebra smooth functions of M . Let us also mention that the starting point
of our approach is quite similar to that of [Hue90] (although metric aspects were
not considered there).
In this note, we show that any Poisson algebra, fulfilling an “almost Kähler condition”, enjoys many properties similar to those of the algebra of smooth functions
on an almost Kähler manifold, opening up for a more metric treatment of Poisson
algebras. Such algebras will be called “Kähler–Poisson algebras”, and we show
that one may associate a Kähler–Poisson algebra to every algebra in a large class
of Poisson algebras. In particular, we prove the existence of a unique Levi-Civita
connection on the module generated by the inner derivations, and show that the
curvature operator has all the classical symmetries. As our approach is quite close
to the theory of Lie-Rinehart algebras, we start by introducing metric Lie-Rinehart
1

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JOAKIM ARNLIND AND AHMED AL-SHUJARY

algebras and recall a few results on the Levi-Civita connection and the corresponding curvature.
In physics, the dynamics of quantum systems are found by using a correspondence between Poisson brackets of functions on the classical manifold, and the commutator of operators in the quantum system. Thus, understanding how properties
of the underlying manifold may be expressed in Poisson algebraic terms enables
both 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 few
the results from [AH14], in order to motivate and understand the introduction of a
Kähler type condition for Poisson algebras, and Section 3 explains how the theory
of Lie-Rinehart algebras can be extended to include metric aspects. In Section 4,
we define Kähler–Poisson algebras and investigate their basic properties as well as
showing that one may associate a Kähler–Poisson algebra to an arbitrary Poisson
algebra in a large class of algebras. In Section 5 we derive a compact formula for the
Levi-Civita connection as well as introducing Ricci and scalar curvature. Section 6
presents a number of examples together with a few detailed computations.
Remark 1.1. We have become aware of the fact that the terminology Kähler–
Poisson structure (resp. Kähler–Poisson manifold ) is used for certain Poisson
structures on a complex manifold where the Poisson bivector is of type (1, 1) (see
e.g. [Kar02]), but we hope that this will not be a source of confusion for the reader.
2. Poisson algebraic formulation of almost Kähler manifolds
In [AH14] it was shown that the geometry of embedded almost Kähler manifolds
can be reformulated entirely in the Poisson algebra of smooth functions. As we shall
develop 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 on
Σ. Furthermore, let us assume that x : (Σ, g) → (Rm , ḡ) is an isometric embedding
of Σ into Rm (with the metric ḡ), and write

p → x(p) = x1 (p), x2 (p), . . . , xm (p) .

The results in [AH14] state that the Riemannian geometry of Σ may be formulated
in terms of the Poisson algebra generated by the embedding coordinates x1 , . . . , xm .
These results hold true as long as there exists a non-zero function γ ∈ C ∞ (Σ) such
that
(2.1)

γ 2 g ab = θap θbq gpq

where θab and gab denote the components of the Poisson bivector and the metric
in local coordinates {ua }na=1 , respectively. If (Σ, ω, g) is an almost Kähler manifold
then 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


gab = ḡ ij ∂a xi ∂b xj ,
and the Poisson bracket is computed as

{f, h} = θab (∂a f )(∂b h).

KÄHLER–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 : Tp Rm → Tp Rm as
D(X) ≡ Di j X j ∂i =

1 i k
{x , x }ḡkl {xj , xl }ḡjm X m ∂i
γ2

for X = X i ∂i ∈ Tp Rm , one computes
1 ab
θ (∂a xi )(∂b xk )ḡkl θpq (∂p xj )(∂q xl )ḡ jm X m
γ2
1
= 2 θab θpq gbq (∂a xi )(∂p xj )ḡ jm X m = g ap (∂a xi )(∂p xj )ḡ jm X m ,
γ

D(X)i =

by using (2.1). Hence, the map D is identified as the orthogonal projection onto
Tp Σ, seen as a subspace of Tp Rm . Having the projection operator at hand, one
may directly proceed to develop the theory of submanifolds. For instance, the
Levi-Civita connection ∇ on Σ is given by

¯ XY
∇X Y = D ∇
¯ is the Levi-Civita connection on (Rm , ḡ). In the
where X, Y ∈ Γ(T Σ) and ∇
particular case (but generically applicable, by Nash’s theorem [Nas56]) when ḡ is
the Euclidean metric, the above formula reduces to
∇X Y i =

1
γ4

m
X

{xi , xk }{xj , xk }X l {xl , xn }{Y j , xn }.

i,j,k,l,n=1

As we intend to develop an analogous theory for Poisson algebras, without any
reference to a manifold, we would like to reformulate (2.1) in terms of Poisson algebraic expressions. Using that gab = ḡij (∂a xi )(∂b xj ) and {xi , xk } = θab (∂a xi )(∂b xj ),
one derives
γ 2 g ab = θap θbq gpq

⇒

γ 2 δca = θap θbq gpq gbc

⇒

γ 2 θar = θap θbq gpq gbc θcr

⇒

γ 2 {xi , xj } = (∂a xi )(∂r xj )θap θbq θcr ḡ kl (∂p xk )(∂q xl )ḡ mn (∂b xm )(∂c xn )

⇒

γ 2 {xi , xj } = −{xi , xk }ḡkl {xl , xn }ḡnm {xm , xj }

which is equivalent to the statement that
(2.2)

γ 2 {f, h} = −{f, xi }ḡij {xj , xk }ḡkl {xl , h}

for all f, h ∈ C ∞ (Σ). Given γ 2 , ḡij and x1 , . . . , xm , the above equation makes
sense in an arbitrary Poisson algebra. The main purpose of this paper is to study
algebras which satisfy such a relation.
3. Metric Lie-Rinehart algebras
The idea of modeling the algebraic structures of differential geometry in a commutative algebra is quite old. We shall follow a pedestrian approach, were we assume
that 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) which
is also a Lie algebra and has an action on A as derivations. Under appropriate assumptions on the ingoing objects, such systems has been studied by many authors
over the years, see e.g [Her53, Koz60, Pal61, Rin63, Nel67, Hue90]. Our starting

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JOAKIM ARNLIND AND AHMED AL-SHUJARY

point is the definition given by G. Rinehart [Rin63]. In the following, we let the
field 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

(3.1)
ω(aα)(b) = a ω(α)(b)

(3.2)
[α, aβ] = a[α, β] + ω(α)(a) β,

for α, β ∈ g and a, b ∈ A. (In most cases, we will leave out ω and write α(a) instead
of 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 derivations of A. It is easy to check that Der(A) is a Lie algebra with respect to composition 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 elements
of Der(A) as derivations.
Example 3.3. Let A = C ∞ (M ) be the algebra (over R) of smooth functions on
a manifold M , and let g = X (A) be the A-module of vector fields on M . With
respect 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 (A1 , g1 ) and (A2 , g2 ) be Lie-Rinehart algebras. A morphism
of Lie-Rinehart algebras is a pair of maps (φ, ψ), with φ : A1 → A2 an algebra
homomorphism and ψ : g1 → g2 a Lie algebra homomorphism, such that


ψ(aα) = φ(a)ψ(α) and φ α(a) = ψ(α) φ(a) ,
for all a ∈ A1 and α ∈ g1 .

A lot of attention has been given to the cohomology of the Chevalley–Eilenberg
complex 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 gk
to an (A, g)-module M , on introduces the standard differential d : C k (g, M ) →
C k+1 (g, M ) as
dτ (α1 , . . . , αk+1 ) =
(3.3)

k+1
X
i=1

+


(−1)i+1 αi τ (α1 , . . . , α̂i , . . . , αk+1 )

k+1
X
i<j


(−1)i+j τ [αi , αj ], α1 , . . . , α̂i , . . . , α̂j , . . . , αk+1 ,

where α̂i indicates that αi is not present among the arguments. The fact that
d ◦ d = 0 implies that one can construct the cohomology of this complex in analogy
with de Rahm cohomology of smooth manifolds. However, as we shall be more
interested in Riemannian aspects, it is natural to study the case when there exists
a metric on the module g. More precisely, we make the following definition.

KÄHLER–POISSON ALGEBRAS

5

Definition 3.5. 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(m1 , m2 ) = g(m2 , m1 ) for all m1 , m2 ∈ M ,
(2) the map ĝ : M → M ∗ , given by ĝ(m1 ) (m2 ) = g(m1 , m2 ), is an A-module
isomorphism,
where M ∗ denotes the dual of M . We shall often refer to property (2) as the metric
being 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 LieRinehart algebras that preserve the metric.
Definition 3.7. Let (A1 , g1 , g1 ) and (A2 , g2 , g2 ) be metric Lie-Rinehart algebras.
A morphism of metric Lie-Rinehart algebras is a morphism of Lie-Rinehart algebras
(φ, ψ) : (A1 , g1 ) → (A2 , g2 ) such that


φ g1 (α, β) = g2 ψ(α), ψ(β)
for all α, β ∈ g1 .

The theory of affine connections can readily be introduced, together with torsionfreeness 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 → EndK (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-module
with connection ∇ and metric g. The connection is called metric if

(3.4)
α g(m1 , m2 ) = g(∇α m1 , m2 ) + g(m1 , ∇α m2 )
for all α ∈ g and m1 , m2 ∈ M .

Definition 3.10. Let (A, g) be a Lie-Rinehart algebra and let ∇ be a connection
on g. The connection is called torsion-free if
∇α β − ∇β α − [α, β] = 0
for all α, β ∈ g.
As in differential geometry, one can show that there exists a unique torsion-free
and metric connection associated to the Riemannian metric. The first step involves
proving Kozul’s formula.
Proposition 3.11. Let (A, g, g) be a metric Lie-Rinehart algebra. If ∇ is a metric
and torsion-free connection on g then it holds that




2g ∇α β, γ = α g(β, γ) + β g(γ, α) − γ g(α, β)
(3.5)
+ g(β, [γ, α]) + g(γ, [α, β]) − g(α, [β, γ])
for all α, β, γ ∈ g.

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Proof. Starting from the right-hand-side of (3.5) and using the metric condition to
rewrite the first three terms as

α g(β, γ) = 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 2g(∇α β, γ).



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 there
exists a unique metric and torsion-free connection on g.
Remark 3.13. 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) = 12 α g(β, γ) + 12 β g(γ, α) − 21 γ g(α, β)
+ 21 g(β, [γ, α]) + 21 g(γ, [α, β]) − 21 g(α, [β, γ]).

By assumption (see Definition 3.5), the metric induces an isomorphism map ĝ :
g → g∗ , which implies that there exists an element ∇α β = ĝ −1 (ω) ∈ g such
that g(∇α β, γ) = ω(γ). This shows that ∇α β exists for all α, β ∈ g such that
relation (3.5) is satisfied. Next, let us show that ∇ defines a connection on g, which
amounts to checking the four properties in Definition 3.8. This is a straight-forward
computation 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 illustrate
the computation with the following example. From (3.5) it follows that



2g(∇aα β, γ) = aα g(β, γ) + β g(γ, aα) − γ g(aα, β)
+ g(β, [γ, aα]) + g(γ, [aα, β]) − g(aα, [β, γ])



= aα g(β, γ) + aβ g(γ, α) + β(a)g(γ, α) − aγ g(α, β) − 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])

= 2ag(∇α β, γ) + β(a)g(γ, α) − γ(a)g(α, β) + γ(a)g(β, α) − β(a)g(γ, α)
= 2ag(∇α β, γ).
The remaining properties of a connection is proved in an analogous way. To show
that ∇ is metric, one again uses (3.5) to substitute g(∇α β, γ) and g(β, ∇α γ) and
find that

α g(β, γ) − 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 and
torsion-free affine connection satisfying (3.5). Moreover, since the metric is nondegenerate, such a connection is unique. Finally, as every metric and torsion-free

KÄHLER–POISSON ALGEBRAS

7

connection on g satisfies (3.5) (by Proposition 3.11) we conclude that there exists
a unique metric and torsion-free connection on g.

In what follows, we shall recall some of the properties satisfied by a metric and
torsion-free connection. The differential geometric proofs goes through with only
a change in notation needed, but we provide them here for easy reference, and
to adapt the formulation to our particular situation. We refer to [Koz60, Nel67]
for a nice overview of differential geometric constructions in modules over general
commutative algebras.
Following the usual definitions, we introduce the curvature as
(3.6)

R(α, β)γ = ∇α ∇β γ − ∇β ∇α γ − ∇[α,β] γ

as well as
R(α, β, γ) = R(α, β)γ
R(α, β, γ, δ) = g(α, R(γ, δ)β).
Let us also consider the extension of ∇ to multilinear maps T : gk → A
k

 X
T α1 , . . . , ∇β αi , . . . , αk ,
(∇β T )(α1 , . . . , αk ) = β T (α1 , . . . , αk ) −
i=1

k

as well as to g-valued multilinear maps T : g → g

k

 X
T α1 , . . . , ∇β αi , . . . , αk .
(∇β T )(α1 , . . . , αk ) = ∇β T (α1 , . . . , αk ) −
i=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-Rinehart
algbera (A, g, g) and let R denote corresponding curvature. Then it holds that
(3.7)
(3.8)

R(α, β, γ) + R(γ, α, β) + R(β, γ, α) = 0,



∇α R (β, γ, δ) + ∇β R (γ, α, δ) + ∇γ R (α, β, δ) = 0,

for all α, β, γ, δ ∈ g.

Proof. The first Bianchi identity (3.7) is proven by acting with ∇γ on the torsion
free condition ∇α β − ∇β α − [α, β] = 0, and then summing over cyclic permutations
of 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 ∇α β −
∇β α − [α, β], γ, δ = 0. One has

0 = R ∇α β − ∇β α − [α, β], γ, δ + 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 yields

0 = ∇γ R(α, β, δ) − ∇γ R (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 second
Bianchi identity follows.


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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-Rinehart
algbera (A, g, g) and let R denote corresponding curvature. Then it holds that
(3.9)

R(α, β, γ, δ) = −R(β, α, γ, δ) = −R(α, β, δ, γ).

(3.10)

R(α, β, γ, δ) = R(δ, γ, α, β),

for all α, β, γ, δ ∈ g.
Proof. The identity R(α, β, γ, δ) = −R(α, β, δ, γ) follows immediately from the definition 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
h
i
h
i
γ g(∇δ α, β) + g(α, ∇δ β) − δ g(∇γ α, β) + g(α, ∇γ β)
− (∇[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 follows
that
(3.11)

R(α, β, γ, δ) + R(α, δ, β, γ) + R(α, γ, δ, β) = 0.

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]).

4. Kähler–Poisson algebras
In this section, we shall introduce a type of Poisson algebras, that resembles the
smooth functions on an (isometrically) embedded almost Kähler manifold, in such
a way that an analogue of Riemannian geometry may be developed. Namely, let
us consider a unital Poisson algebra (A, {·, ·}) and let {x1 , . . . , xm } be a set of
distinguished elements of A, corresponding to functions providing an embedding
into Rm , in the geometrical case. One may also consider the setting of algebraic
(Poisson) varieties where A is a finitely generated Poisson algebra and {x1 , . . . , xm }
denotes a set of generators. Our aim is to introduce equation (2.2) in A and
investigate just how far one may take the analogy with Riemannian geometry. After
introducing Kähler–Poisson algebras below, we will show that they are, in a natural
way, metric Lie-Rinehart algebras, which implies that the results of Section 3 can
be applied; in particular, there exists a unique torsion-free metric connection on
every Kähler–Poisson algebra. Note that Lie-Rinehart algebras related to Poisson
algebras 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 Kähler
manifold:
(2.2)

γ 2 {f, h} = −{f, xi }ḡij {xj , xk }ḡ kl {xl , h}.

KÄHLER–POISSON ALGEBRAS

9

This equation is well-defined in a Poisson algebra, and we shall use it to define the
main object of our investigation.
Definition 4.1. Let A be a Poisson algebra over K and let {x1 , . . . , xm } ⊆ A.
Given gij ∈ A, for i, j = 1, . . . , m, such that gij = gji , we say that the triple
K = A, {x1 , . . . , xm }, g is a Kähler–Poisson-algebra if there exists η ∈ A such
that
m
X
(4.1)
η{a, xi }gij {xj , xk }gkl {xl , b} = −{a, b}
i,j,k,l=1

for all a, b ∈ A.

Remark 4.2. From now on, we shall use the differential geometric convention that
repeated indices are summed over from 1 to m, and omit explicit summation symbols.
Given a Kähler–Poisson-algebra K, we let g denote the A-module generated by all
inner derivations, i.e.
g = {a1 {c1 , ·} + · · · + aN {cN , ·} : ai , ci ∈ A and N ∈ N}.
It is a standard fact that g is a Lie algebra over K with respect to


[α, β](a) = α β(a) − β α(a) .

The matrix g induces a bilinear symmetric form on g, defined by
(4.2)

g(α, β) = α(xi )gij β(xj ),

and we refer to g as the metric on g. To the metric g one may associate the map
ĝ : g → g∗ defined as
ĝ(α)(β) = g(α, β).

Proposition 4.3. If K = A, {x1 , . . . , xm }, g is a Kähler–Poisson-algebra then
the metric g is non-degenerate; i.e. the map ĝ : g → g∗ is a module isomorphism.
Proof. Let us first show that g is injective; i.e. we will show that ĝ(α)(β) = 0,
for all β ∈ g, implies that α = 0. Thus, write α = αi {xi , ·}, and assume that
g(α, β) = 0 for all β ∈ g. In particular, we can choose β = η{c, xk }gkm {·, xm }, for
arbitrary c ∈ A, which implies that
0 = g(α, β) = ηαk {xk , xi }gij {c, xk }gkm {xj , xm }
= −αk η{xk , xi }gij {xj , xm }gmk {xk , c}.
Using the relation (4.1), one obtains
αk {xk , c} = 0
for all c ∈ A, which is equivalent to α = 0. This shows that ĝ is injective. Let us
now show that ĝ is surjective. Thus, let ω ∈ g∗ and set
α = ηω({xi , ·})gij {xj , ·} ∈ g,
which gives
ĝ(α)(ak {bk , ·}) = ηω({xi , ·})gij {xj , xl }glm ak {bk , xm }
= −ηak {bk , xm }gml {xl , xj }gji ω({xi , ·}).

10

JOAKIM ARNLIND AND AHMED AL-SHUJARY

Since ω is a module homomorphism one obtains
ĝ(α)(ak {bk , ·}) = ω(−ηak {bk , xm }gml {xl , xj }gji {xi , ·})
= ω(ak {bk , ·}),
by using (4.1), which proves that every element of g∗ is in the image of ĝ. We
conclude that ĝ is a module isomorphism.

Corollary 4.4. If (A, {x1 , . . . , xm }, g) is a Kähler–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-Rinehart
algebra, and Proposition 4.3 implies that the metric is non-degenerate. Hence,
(A, g, g) is a metric Lie-Rinehart algebra.

Let us now introduce some notation for Kähler–Poisson algebras. Thus, we set
P ij = {xi , xj }
P i (a) = {xi , a},
for a ∈ A, as well as
Dij = ηP i k P jk = η{xi , xl }glk {xj , xk }
Di (a) = ηP k (a)Pk i = η{xk , a}gkl {xl , xi },
and note that Dij = Dji . With respect to this notation, (4.1) can be stated as
Di (a)Pi (b) = {a, b}.

(4.3)

The metric will be used to lower indices in analogy with differential geometry. E.g.
P i j = P ik gkj

Di j = Dik gkj .

Furthermore, one immediately derives the following useful identities
(4.4)

Dij Pj (a) = P i (a),

P ij Dj (a) = P i (a) and Di j Djk = Dik .

by using (4.1).
There is a natural embedding ι : g → Am , given by
ι(ai {bi , ·}) = ai {bi , xk }ek ,
m
where {ek }m
k=1 denotes the canonical basis of the free module A . Moreover, g
m
defines a bilinear form on A via

g(X, Y ) = X i gij Y j
for X = X i ei ∈ Am and Y = Y i ei ∈ Am , and we introduce the map D : Am → Am
by setting
D(X) = Di j X j ei
for X = X i ei ∈ Am .
Proposition 4.5. The map D : Am → Am is an orthogonal projection; i.e.
D2 (X) = D(X) and g(D(X), Y ) = g(X, D(Y ))
for all X, Y ∈ Am .

KÄHLER–POISSON ALGEBRAS

11

Proof. First, it is clear that D is an endomorphism of Am . It follows immediately
from (4.4) that
D2 (X) = Di j Dj k X k ei = Di j Djl glk X k ei = Dil glk X k ei = Di k X k ei = D(X).
Furthermore, using that Dij = Dji one finds that

g D(X), Y = Di j X j gik Y k = X j Dil glj gik Y k = X j glj Dli gik Y k

= X j gjl Dl k Y k = g X, D(Y ) ,
which completes the proof.



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 finitely
generated projective module by showing that g is isomorphic to T A.
Proposition 4.6. The map ι : g → Am 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 Am , elements of T A can be characterized by the fact
that D(X) = X for all X ∈ T A. Thus, by showing that

D ι(ak {bk , ·}) = Di j ak {bk , xj }ei = −ak Di j P j (bk )ak ei
= −ak P i (bk ) = ι(ak {bk , ·})

it follows that ι(ak {bk , ·}) ∈ T A. Let us now show that ι is injective; assume that
ι(ak {bk , ·}) = 0, which implies that
ak {bk , xi } = 0

for i = 1, . . . , m.

Next, for arbitrary c ∈ A, we write
ak {bk , c} = −ηak {bk , xi }gij P jl glm {xm , c},
by using (4.1). Since ak {bk , xi } = 0, one obtains ak {bk , c} = 0 for all c ∈ A.
To prove that ι is surjective, we start from an arbitrary X = X i ei ∈ T A, and
note that

ι X i gij Di (·) = X i gij Dik ek = D(X) = X

by using that D(X) = X for all X ∈ T A. Hence, we may conclude that ι is an
isomorphism from g to T A.

Corollary 4.7. g is a finitely generated projective module.
Note that the above result is clearly not dependent on whether or not the underlying Poisson algebra has the structure of a Kähler–Poisson algebra, as the definition
of g involves only inner derivations. Hence, as soon as the Poisson algebra admits the structure of a Kähler–Poisson algebra, it follows that the module of inner
derivations is projective. Furthermore, the fact that g is a projective module has
several implications for the underlying Lie-Rinehart algebra [Rin63, Hue90]. Next,
let us show that the derivations Di generate g as an A-module.
Proposition 4.8. The A-module g is generated by {D1 , . . . , Dm }.

12

JOAKIM ARNLIND AND AHMED AL-SHUJARY

Proof. First of all, it is clear that every element in the module generated by Di ,
written as
α(c) = αi Di (c) = ηαi {xi , xj }gjk {c, xk },
is an element of g. Conversely, let α ∈ g be an arbitrary element written as
X
α(c) =
aN {bN , c}.
N

for c ∈ A. Using the Kähler–Poisson condition (4.1) one may write this as
X
X
α(a) =
aN {bN , c} = −
ηaN {bN , xi }gij {xj , xk }gkl {xl , c}
N

=

X

N

N

i



aN {b , x }gij Dj (c),

N

which clearly lies in the module generated by {D1 , . . . , Dm }.



Thus, every α ∈ g may be written as α = αi Di for some αi ∈ A. It turns out that
this is a very convenient way of writing elements of g, which shall be extensively
used in the following. Note that if the Kähler–Poisson algebra comes from an almost
Kähler manifold M , then Di is quite close to a partial derivative on M in the sense
that (∂a xi )gik Dk (f ) = ∂a f , for f ∈ C ∞ (M ).
4.1. The trace of linear maps. As we shall be interested in both Ricci and scalar
curvature, which are defined using traces of linear maps, we introduce

(4.5)
tr(L) = g L(Di ), Dj Dij .

for an A-linear map L : g → g. This trace coincides with the ordinary trace on
g∗ ⊗A g; namely, consider
X
L=
ωN ⊗A αN ∈ g∗ ⊗A g
N

as a linear map L : g → g in the standard way via
X
L(β) =
ωN (β)αN ,
N

together with
tr(L) =

X

ωN (αN ).

N

i
Writing αN = αN
i D one finds that
X
X


k
j
kj
g L(Di ), Dj Dij =
g ωN (Di )αN
ωN (Di )αN
k D , D Dij =
k D Dij
N

=

X
N

k
i
ωN (αN
k D iD )

=

X
N

N

k
ωN (αN
k D )

=

X

ωN (αN ).

N

In particular, this implies that the trace defined via (4.5) is independent of the
Kähler–Poisson structure.

KÄHLER–POISSON ALGEBRAS

13

4.2. Morphisms of Kähler–Poisson algebras. As Kähler–Poisson algebras are
also metric Lie-Rinehart algebras, we shall require that a morphism of Kähler–
Poisson algebras is also a morphism of metric Lie-Rinehart algebras (as defined
in Section 3). However, as the definition of a Kähler–Poisson also involves the
choice of a set of distinguished elements, we will require a morphism to respect
the subalgebra generated by these elements. To this end, we start by making the
following definition.
Definition 4.9. Given a Kähler–Poisson algebra (A, {x1 , . . . , xm }, g), let Afin ⊆ A
denote the subalgebra generated by {x1 , . . . , xm }.
Equipped with this definition, we introduce morphisms of Kähler–Poisson algebras
in the following way.
′

Definition 4.10. Let K = (A, {x1 , . . . , xm }, g) and K′ = (A′ , {y 1 , . . . , y m }, g ′ ) be
Kähler–Poisson algebras together with their corresponding modules of derivations
g and g′ , respectively. A morphism of Kähler–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 the
metric Lie-Rinehart algebras (A, g, g) and (A, g′ , g ′ ) and φ is a Poisson algebra
homomorphism such that φ(Afin ) ⊆ A′fin .
Note that if the algebras are finitely generated such that A = Afin and A′ = A′fin
(which is the case in many examples), the condition φ(Afin ) ⊆ A′fin is automatically
satisfied. Although a morphism of Kähler–Poisson algebras is given by a choice of
two maps φ and ψ, it is often the case that φ determines ψ in the following sense.
′

Proposition 4.11. Let (φ, ψ) : (A, {x1 , . . . , xm }, g) → (A′ , {y 1 , . . . , y m }, g ′ ) be a
morphism of Kähler–Poisson algebras such that for all α′ ∈ g′

α′ φ(a) = 0 ∀ a ∈ A ⇒ α′ = 0
then

Proof. Let


ψ a{b, ·}A = φ(a){φ(b), ·}A′ .

′

(φ, ψ) : (A, {x1 , . . . , xm }, g) → (A′ , {y 1 , . . . , y m }, g ′ )
be a morphism of Kähler–Poisson algebras fulfilling the assumption above. Since φ
is a Poisson algebra homomorphism, one obtains for α = a{b, ·}A


φ α(c) = φ a{b, c}A = 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. 
For instance, the requirements in Proposition 4.11 are clearly satisfied if φ is surjective.

14

JOAKIM ARNLIND AND AHMED AL-SHUJARY

4.3. Construction of Kähler–Poisson algebras. Given a Poisson algebra (A, {·, ·})
one may ask if there exist {x1 , . . . , xm } and gij such that (A, {x1 , . . . , xm }, g) is a
Kähler–Poisson algebra? Let us consider the case when A is a finitely generated
algebra, and let {x1 , . . . , xm } be an arbitrary set of generators. If we denote by P
the matrix with entries {xi , xj } and by g the matrix with entries gij , the Kähler–
Poisson condition (4.1) may be written in matrix notation as
ηPgPgP = −P.
Given an arbitrary antisymmetric matrix P, we shall find g by first writing P in
a block diagonal form, with antisymmetric 2 × 2 matrices on the diagonal. This is
a well known result in linear algebra, in which case the eigenvalues appear in the
diagonal blocks. For an antisymmetric matrix with entries in a commutative ring,
a similar result holds.
Lemma 4.12. Let MN (R) denote the set of N × N matrices with entries in R. For
N ≥ 2, let P ∈ MN (R) be an antisymmetric matric. Then there exists V ∈ MN (R),
an antisymmetric Q ∈ MN −2 (R) and λ ∈ R such that


0 λ
0
.
V T PV =  −λ 0
0
Q

Proof. We shall construct the matrix V by using elementary row and column operations. Note that if a matrix E represents an elementary row operation, then
E T PE is obtained by applying the elementary operation to both the row and the
corresponding column. Denoting the matrix elements of P by pij , we start by constructing a matrix Vk such that (VkT PVk )k1 = (VkT PVk )k2 = 0 (which necessarily
implies that also the (1k) and (2k) matrix elements are zero). To this end, let Vk1
denote the matrix representing the elementary row operation that multiplies the
k’th row by p12 , and let Vk2 represent the operation that adds the first row, multiplied by −pk2 , to the k’th row. Furthermore, Vk3 represents the operation of adding
the second row, multiplied by pk1 , to the k’th row. Setting Vk = Vk1 Vk2 Vk3 it is easy
to see that VkT PVk is an antisymmetric matrix where the (1k), (2k), (k1) and (k2)
matrix elements are zero. Consequently, we set V = V3 V4 · · · VN and conclude that
V T PV is of the desired form.

Proposition 4.13. Let P ∈ MN (R) be an antisymmetric matric, and let N̂ denote
the integer part of N/2. Then there exists V ∈ MN (R) and λ1 , . . . , λN̂ ∈ R such
that
V T PV = diag(Λ1 , . . . , ΛN̂ )

if N is even,

T

V PV = diag(Λ1 , . . . , ΛN̂ , 0)
where
Λk =



0
−λk

if N is odd,


λk
.
0

Proof. Let us prove the statement by using induction together with Lemma 4.12.
Thus, assume that there exists V ∈ MN (R) such that
V T PV = diag(Λ1 , . . . , Λk , Qk+1 )
where Qk+1 ∈ MN −2k is an antisymmetric matrix. Clearly, by Lemma 4.12, this
holds true for k = 1. Next, assume that N − 2k ≥ 2. Applying Lemma 4.12 to

KÄHLER–POISSON ALGEBRAS

15

T
Qk+1 we conclude that there exists Vk+1 ∈ MN −2k (R) such that Vk+1
Qk+1 Vk+1 =
diag(Λk+1 , Qk+2 ). Furthermore, defining Wk+1 ∈ MN (R) by Wk+1 = diag(12k , Vk+1 )
one finds that

(V Wk+1 )T P(V Wk+1 ) = diag(Λ1 , . . . , Λk+1 , Qk+2 ).
By induction, it follows that one may repeat this procedure until N − 2k < 2. If N
is even, then N − 2k = 0 and the statement follows. If N is odd, then N − 2k = 1
and, since V T PV is antisymmetric, it follows that the (N N ) matrix element is
zero, giving the stated result.

Returning to the case of a Poisson algebra generated by x1 , . . . , xm , assume for the
moment that m = 2N for a positive integer N . By Proposition 4.13, there exists a
matrix V
V T PV = P0
where P0 is a block diagonal matrix of the form
P0 = diag(Λ1 , . . . , ΛN )
with
Λk =



0
−λk


λk
.
0

In the same way, defining g0 = diag(g1 , . . . , gN ) with


λ 1 0
gk =
λk 0 1
λ = λ1 · · · λN
we set g = V g0 V T . Noting that
P0 g0 P0 g0 P0 = −λ2 P0
one finds
0 = P0 g0 P0 g0 P0 + λ2 P0 = V T PV g0 V T P0 V gV T P0 V + λ2 V T PV

= V T PgPgP + λ2 P V

It is a general fact that for an arbitrary matrix V there exists a matrix Ṽ such that
Ṽ V = V Ṽ = (det V )1. Multiplying the above equation from the left by Ṽ T and
from the right by Ṽ yields

(4.6)
det(V )2 PgPgP + λ2 P = 0.
As long as det(V ) is not a zero divisor, this implies that
PgPgP = −λ2 P.
Thus, given a finitely generated Poisson algebra A, the above procedure gives a
rather general way to associate a localization A[λ−1 ] and a metric g to A, such
that (A[λ−1 ], {x1 , . . . , xm }, g) is a Kähler–Poisson algebra. Note that the above
argument, with only slight notational changes, also applies to the case when m is
odd, in which case an extra block of 0 will appear in P0 .

16

JOAKIM ARNLIND AND AHMED AL-SHUJARY

5. The Levi-Civita connection
Since every Kähler–Poisson algebra is also a metric Lie-Rinehart algebra, the results
of Section 3 immediately applies. In particular, there exists a unique torsion-free
and metric connection on the module g. In this section, we shall derive an explicit
expression for the Levi-Civita connection of an arbitrary Kähler–Poisson algebra.
It turns out to be convenient to formulate the results in terms of the generators
{D1 , . . . , Dm }. Kozul’s formula gives the connection as



2g(∇Di Dj , Dk ) = Di g(Dj , Dk ) + Dj g(Dk , Di ) − Dk g(Di , Dj )
(5.1)
− g([Dj , Dk ], Di ) + g([Dk , Di ], Dj ) + g([Di , Dj ], Dk ),
and one notes that an element α = a{b, ·} ∈ g may be recovered from g(α, Di ) as
g(α, Di )Di (f ) = a{b, xk }Di k Di (f ) = a{b, xk }Dk (f ) = a{b, f } = α(f ).
Thus, one immediately obtains ∇Di Dj = g(∇Di Dj , Dk )Dk . However, it turns out
that one can obtain a more compact formula for the connection. Let us start by
proving the following result.


Lemma 5.1. g([Di , Dj ], Dk ) = Di Djk − Dj Dik .

Proof. For convenience, let us introduce the notation P̂ ij = η{xi , xj } and, consequently, P̂ i j = P̂ ik gkj . In this notation, one finds Di (a) = P̂ i j {a, xj }. Thus, one
obtains
g([Di , Dj ], Dk ) = [Di , Dj ](xl )Dk l = P̂ i m {Djl , xm }Dk l − P̂ j n {Dil , xn }Dk l

= P̂ i m {P̂ j n {xl , xn }, xm } − P̂ j n {P̂ i m {xl , xm }, xn } Dk l

= P̂ i m P̂ j n − {{xn , xl }, xm } − {{xl , xm }, xn } Dk l

+ P̂ i m {P̂ j n , xm }{xl , xn } − P̂ j n {P̂ i m , xn }{xl , xm } Dk l
= P̂ i m P̂ j n {{xm , xn }, xk } + P̂ i m {P̂ j n , xm }{xk , xn }
− P̂ j n {P̂ i m , xn }{xk , xm },
by using the Jacobi identity together with {a, xi }Dk i = {a, xk }. Furthermore, in
the second and third term, one uses Leibniz’s rule to obtain
g([Di , Dj ], Dk ) = P̂ i m P̂ j n {{xm , xn }, xk } + P̂ i m {P̂ j n {xk , xn }, xm }
− P̂ i m P̂ j n {{xk , xn }, xm } − P̂ j n {P̂ i m {xk , xm }, xn } + P̂ j n P̂ i m {{xk , xm }, xn }

= P̂ i m P̂ j n {{xm , xn }, xk } + {{xn , xk }, xm } + {{xk , xm }, xn }


+ Di Djk − Dj (Dik ) = Di Djk − Dj (Dik ),

by again using the Jacobi identity.



The above result allows for the following formulation of the Levi-Civita connection
for a Kähler–Poisson algebra.
Proposition 5.2. If ∇ denotes the Levi-Civita connection of a Kähler–Poisson
algebra K then
1
1
1
(5.2)
∇Di Dj = Di (Djk )Dk − Dj (Dik )Dk + Dk (Dij )Dk ,
2
2
2

KÄHLER–POISSON ALGEBRAS

17

or, equivalently, ∇Di Dj = Γij k Dk where
(5.3)

Γij k =

1
1
1 i jl
D (D )Dlk − Dj (Dil )Dlk + Dk (Dij ).
2
2
2

Proof. Since g(Di , Dj ) = Dij , Kozul’s formula (5.1) together with Lemma 5.1 gives
2g(∇Di Dj , Dk ) = Di (Djk ) + Dj (Dki ) − Dk (Dij ) − Dj (Dki ) + Dk (Dji )
+ Dk (Dij ) − Di (Dkj ) + Di (Djk ) − Dj (Dik )
= Di (Djk ) − Dj (Dki ) + Dk (Dij ),
which proves (5.2). The fact that one may write the connection as ∇Di Dj = Γij k Dk
follows from Dij Dj = Di and Dk (a)Dk (b) = Dk (a)Dk (b).

Thus, for arbitrary elements of g, one obtains
∇α β = α(βi )Di + Γij k αi βj Dk

(5.4)

where α = αi Di and β = βi Di , and curvature is readily introduced as
R(α, β)γ = ∇α ∇β γ − ∇β ∇α γ − ∇[α,β] γ.
Ricci curvature is defined as
Ric(α, β) = tr γ → R(γ, α)β
and using the trace from Section 4.1, one obtains



Ric(α, β) = g(R(Di , α)β, Dj )Dij .
To define the scalar curvature, one considers the Ricci curvature as a linear map
Ric : g → g

with

Ric(α) = Ric(α, Di )Di ,

giving


S = tr α → Ric(α) = g R(Di , Dk )Dl , Dj Dij Dkl .

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 is
easy to see that
∇f = Di (f )Di
since
g(Di (f )Di , αj Dj ) = Di (f )αj Dij = αj Dj (f ) = α(f ).
The divergence of an element α ∈ g is defined as
div(α) = tr(β → ∇β α),
and, finally, the Laplacian
∆(f ) = div(∇f ).

18

JOAKIM ARNLIND AND AHMED AL-SHUJARY

6. Examples
As shown in Section 2, the algebra of smooth functions on an almost Kähler manifold M becomes a Kähler–Poisson algebra when choosing x1 , . . . , xm to be embedding coordinates, providing an isometric embedding into Rm , endowed with
the standard Euclidean metric. (Recall that, by Nash’s theorem [Nas56], such an
embedding always exists.) In this section, we shall present examples of a more
algebraic nature to illustrate the fact that algebras of smooth functions are not the
only examples of Kähler–Poisson algebras.
Keeping in mind the general construction procedure in Section 4.3, we consider
finitely generated Poisson algebras with a low number of generators.
6.1. Poisson algebras generated by two elements. Let A be a unital Poisson
algebra generated by the two elements x1 = x ∈ A and x2 = y ∈ A, and set


0
{x, y}
P=
−{x, y}
0
It is easy to check that for an arbitrary symmetric matrix g
PgPgP = −{x, y}2 det(g)P.
Thus, as long as {x, y}2 det(g) is not a zero-divisor, one may localize to obtain a
Kähler–Poisson algebra
K = (A[({x, y}2 det(g))−1 ], {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 A0 , generated by two
elements. Let us start by choosing an element λ ∈ A0 for which the localization
A = A0 [p−1 , λ−1 ] exists, and then defining the metric as


1 1 0
g=
λ 0 1
From the above considerations, we know that (A, {x, y}, g) is a Kähler–Poisson
algebra with η = λ2 /p2 , where p = {x, y}. For convenience we also introduce
γ = p/λ such that η = 1/γ 2. Let us start by computing the derivations Dx = D1
and Dy = D2 , which generate the module g:
λ
1
{·, y} = − {y, ·}
p
γ
λ
1
Dy = η{y, xi }gij {·, xj } = − {·, x} = {x, ·}
p
γ

Dx = η{x, xi }gij {·, xj } =

as well as
1 x
1
D
and Dy = g2k Dk = Dy .
λ
λ
Moreover, they provide an orthogonal set of generators since
Dx = g1k Dk =

1
1
1 p2
{y, xi }gij {y, xj } = 2
=λ
γ
γ
γ λ
g(Dy , Dy ) = λ
g(Dx , Dy ) = 0,
g(Dx , Dx ) =

KÄHLER–POISSON ALGEBRAS

and one obtains

(Dij ) = g(Di , Dj ) =

19



λ 0
.
0 λ

Note that g is a free module with basis {Dx , Dy } since
(
(
x
y
−a γ1 {y, x} = 0
aD
(x)
+
bD
(x)
=
0
aDx + bDy = 0 ⇒
⇒
b γ1 {x, y} = 0
aDx (y) + bDy (y) = 0

⇒

(

a=0
b=0

by using that λ is invertible.
Let us introduce the derivation Dλ = γ −1 {λ, ·} and note that
1 x
1
D (λ)Dy − Dy (λ)Dx .
λ
λ
From Proposition 5.2 one computes the connection:
1
1
1
∇Dx Dx = D1 (D1k )Dk − D1 (D1k )Dk + Dk (D11 )Dk
2
2
2
1 y
1 i
1 x
= D (λ)Dx + D (λ)Dy = D (λ)Di
2
2
2
and similarly
1
1
∇Dy Dy = Dx (λ)Dx + Dy (λ)Dy = ∇Dx Dx
2
2
1 y
1 x
y
∇Dx D = D (λ)Dy − D (λ)Dx = Dλ
2
2
1
1 y
x
∇Dy D = D (λ)Dx − Dx (λ)Dy = −Dλ
2
2
Moreover, the curvature can readily be computed


 1
 y
1
x
y
x
2
2
x
y
R(D , D )D = Dx (λ) + Dy (λ) − Dx D (λ) − Dy D (λ) D
2
2



 x
1
1
x
y
y
2
2
x
y
R(D , D )D = − Dx (λ) + Dy (λ) − Dx D (λ) − Dy D (λ) D ,
2
2
Dλ = [Dx , Dy ] =

as well as the scalar curvature



1
S=
Dx Dx (λ) + Dy Dy (λ) − 2Dx (λ)2 − 2Dy (λ)2 .
λ
Moreover, one finds that
∇f = Dx (f )Dx + Dy (f )Dy

div(αx Dx + αy Dy ) = Dx (αx ) + Dy (αy )


∆(f ) = Dx Dx (f ) + Dy Dy (f )


= Dx Dx (f ) + Dy Dy (f ) − Dx (λ)Dx (f ) − Dy (λ)Dy (f ).

6.2. Poisson algebras generated by three elements. Let A be a unital Poisson
algebra generated by x1 = x, x2 = y, x3 = z ∈ A. Writing {x, y} = a, {y, z} = b
and {z, x} = c, i.e.


0
a −c
b ,
P = −a 0
c −b 0

20

JOAKIM ARNLIND AND AHMED AL-SHUJARY

one readily checks that for an arbitrary symmetric matrix g
PgPgP = −τ P
with
τ = a2 |g|33 + b2 |g|11 + c2 |g|22 + 2ab|g|31 − 2ac|g|32 − 2bc|g|21 ,
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 Kähler–Poisson algebra
K = {A[τ −1 ], {x, y, z}, g}.
In particular, if g = diag(λ, λ, λ), then τ = λ2 (a2 + b2 + c2 ).
Let us now construct a particular class of algebras with a natural geometric
interpretation and a close connection to algebraic geometry. Let R[x, y, z] be the
polynomial ring in three variables over the real numbers, and write x1 = x, x2 = y
and x3 = z. For arbitrary C ∈ R[x, y, z], it is straight-forward to show that
{xi , xj } = εijk ∂k C,
where εijk denotes the totally antisymmetric symbol with ε123 = 1, defines a Poisson
structure on R[x, y, z] which is well-defined on the quotient AC = R[x, y, z]/(C)
since
{xi , C} = {xi , xj }∂j C = εijk (∂k C)(∂j C) = 0.
In the spirit of algebraic geometry, the algebra AC has a natural interpretation
as the polynomial functions on the level set C(x, y, z) = 0 in R3 . Choosing the
metric gij = δij 1 (corresponding to the Euclidean metric on R3 ) one obtains a
Kähler–Poisson algebra (AbC , {x, y, z}, g) where
2
2
2
AbC = AC [τ −1 ] and τ = ∂x C + ∂y C + ∂z C ,

with η = τ −1 . Note that the points in R3 , for which τ (x, y, z) = 0, coincide with
the 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 = 21 (ax2 + by 2 + cz 2 − 1) for a, b, c ∈ R, giving
{x, y} = cz,

{y, z} = ax

and

and
η = a2 x2 + b2 y 2 + c2 z 2
together with



b 2 y 2 + c2 z 2
ij

−abxy
(D ) = η
−acxz

−abxy
a2 x2 + c2 z 2
−bcyz

{z, x} = by.
−1


−acxz
−bcyz  .
2 2
a x + b2 y 2

A straight-forward, but somewhat lengthy, calculation gives
 x
 x
 x
 x
D
D
D
D
b D y 
b D y 
R(Dy , Dz ) Dy  = ax R
R(Dx , Dy ) Dy  = cz R
Dz
Dz
Dz
Dz


 x
 x
0
−cz by
D
D
b = abcη 3  cz
b Dy  where R
0
−ax ,
R(Dz , Dx ) Dy  = by R
z
z
−by ax
0
D
D

KÄHLER–POISSON ALGEBRAS

21

and the scalar curvature becomes
S = 2abcη 2 .
7. Summary
In this note, we have introduced the concept of Kähler–Poisson algebras as a mean
to 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 the
algebra, which share crucial properties with their classical counterparts. The idea
behind the construction was to identify a distinguished set of elements in the algebra
that serve as “embedding coordinates”, and then construct the projection operator
D that projects from the tangent space of the ambient manifold onto that of the
embedded submanifold. It is somewhat surprising that (4.1) encodes the crucial
elements that are needed for the algebra to resemble an algebra of functions on an
almost Kähler manifold.
As outlined in Section 4.3, a large class of Poisson algebras admit a Kähler–
Poisson algebra as an associated localization, which shows a certain generality of
our treatment. Thus, even if one is not interested in metric structures on a Poisson
algebra, the tools we have developed might be of help. For instance, if a Poisson
algebra can be given the structure of a Kähler–Poisson algebra, one immediately
concludes that the module generated by the inner derivations is a finitely generated
projective module. A statement which is clearly independent of any metric structure. A comparison with differential geometry is close at hand, where the structure
of a Riemannian manifold can be used to prove results about the underlying manifold (or even the topological structure).
Let us end with a brief outlook. After having studied the basic properties of
Kähler–Poisson algebras in this paper, there are several natural questions that
can be studied. For instance, what is the interplay between the cohomology (of
Lie-Rinehart algebras) and the Levi-Civita connection? Can one perhaps use the
connection to compute cohomology? Is there a natural way to study the moduli
spaces of Poisson algebras; i.e. how many (non-isomorphic) Kähler–Poisson structures does there exist on a given Poisson algebra? We hope to return to these, and
many 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.
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(Joakim Arnlind) Dept. of Math., Linköping University, 581 83 Linköping, Sweden
E-mail address: joakim.arnlind@liu.se
(Ahmed Al-Shujary) Dept. of Math., Linköping University, 581 83 Linköping, Sweden
E-mail address: ahmed.al-shujary@liu.se

