Journal of Applied Mathematics and Physics
Vol.06 No.03(2018), Article ID:83227,24 pages
10.4236/jamp.2018.63048
Poisson (Co)homology of a Class of Frobenius Poisson Algbras
Mengyao Wang
College of Science, University of Shanghai for Science and Technology, Shanghai, China
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 5, 2018; Accepted: March 20, 2018; Published: March 23, 2018
ABSTRACT
In this paper, we study the truncated polynomial algebra L in n variables, and discuss the following four problems in detail: 1) Homology complex and homology group of Poisson algebra L; 2) Given a new Poisson bracket by calculation modular derivation of Frobenius Poisson algebra; 3) Calculate the twisted homology group of Poisson algebra L; 4) Verify the theorem of twisted Poincaré duality between twisted Poisson homology and Poisson Cohomology.
Keywords:
Poisson Algebra, Poisson (Co)homology, Twisted Poisson Module, Twisted Poincaré Duality
1. Introduction
Poisson structures appear in a large variety of different contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. They play an important role in Poisson geometry, in algebraic geometry and non-commutative. Poisson cohomologies are important invariants of Poisson structures. The set of Casimir elements of the Poisson structure is the 0-th cohomology; Poisson derivations modulo Hamiltonian derivations are the 1-st cohomology. Poisson cohomology appears as one considers deformations of Poisson algebras. Given a Poisson algebra, we can get some vital information about the Poisson algebra structure from calculate its Poisson (Co)homology.
C. Kassel started calculate the (Co)homology of linear Poisson structure (see [1] ). Luo J, Wang S.Q (see [2] ) get the Twisted Poincare duality between Poisson homology and Poisson cohomology in quadratic Poisson algebra. Roger C and Vanhaecke P (see [3] ) do the research about the Poisson cohomology of the affine plane. Pichereau calculate the Poisson homology in 3 dimension affine space (see [4] ). Tagne Pelap calculates the Poisson (Co)homology of polynomial algebra with 3 variable in generalized Jacobian Poisson structure (see [5] ). Foramlity theorem has been proved by Kontsevich in 2003 (see [6] ), and the results revealed the importance Poisson algebra and its deformation quantization. In general, it is very important to calculate Poisson cohomology from a given Poisson structure. These researches mainly focused on the smooth algebra and the finite dimension algebra.
The homology theorem of the singular algebra is few. Launois S and Richard L [7] calculate the Poisson (Co)homology of truncated polynomial algebras in 2 variables, and established the twisted Poincaré duality between them. [8] and [9] proofed this conclusion stands for all Frobenius Poisson algebra. In this paper, we want to study infinite dimension situation: a truncated polynomial algebra with n variables.
2. Main Results
In this paper, we let k is a number field. We consider the truncated polynomial algebras in n variables
with the Poisson bracket
We get some mainly results. In part 4, we get the i-th homology group
of algebra L
,
.
In part 5, we calculate Poisson modular derivation after we get the Frobenius pairing
then we can get the new Poisson module structure
In part 6, we get the twisted Poisson homology group and we get the results
,
the elements in
with the length of
in
.
In part 7, we check the twisted Poincaré duality between Poisson homology and Poisson Cohomology
by calculating the Twisted Poisson cohomology.
3. Some Preliminary Definition and Proposition
Definition 1 [10] . A Poisson algebra is an k-vector space A equipped with two multiplications
and
such that
1)
is a commutative associative algebra over k, with unit 1;
2)
is a Lie algebra over F;
3) The two multiplications are compatible in the sense that
where x, y and z are arbitrary elements of A. The Lie bracket
is then called a Poisson bracket
Example 1. The commutative polynomial algebra in two variables
is a Poisson algebra for the bracket defined on the generators by
or equivalently for any
More generally, for any
,
is a Poisson algebra for the “symplectic” bracket defined on the generators by
, for all
or equivalently for any
We refer to this Poisson algebra as the Poisson-Weyl algebra, denoted by
.
Example 2. For any
, the commutative polynomial algebra
is a Poisson algebra for the “multiplicative” bracket defined on the generators by
More generally, for any
and for any
antisymmetric matrix
with entries in k,
is a Poisson algebra for the bracket defined on the generators by
for all
.
We refer to this Poisson algebras as the Poisson-quantum plane and Poisson-quantum space respectively, denoted by
and
.
Definition 2 [11] . Let
be a Poisson algebra. A right A-module
is called a right Poisson module over A, if there is a bilinear map
such that
1)
is a right Lie-module over the Lie algebra
;
2)
for any
;
3)
for any
.
Left Poisson modules are defined similarly. In particular, any Poisson algebra R is naturally a right and left Poisson module over itself
For a Poisson algebra A, the space A has a natural right (also, left) Poisson module structure. Given two right Poisson modules
and
over A, a k-linear map
is called a morphism of Poisson modules if
for each
and
. The following two properties on Poisson modules are straightforward.
Proposition 1 [8] . Suppose that
is a right Poisson module over A, Then the following actions define a left Poisson module structure on
for each
. Similarly, a left Poisson module M yields a right Poisson module
.
Proposition 2 [8] . Let
be a right Poisson module over A and M be a right A-module. Suppose that
is an isomorphism of A-modules. Then there exists a right Poisson module structure on M given by:
. for each
, such that f is an isomorphism of Poisson modules.
Definition 3 [2] . Let R be a Poisson algebra. In general, let
be the Kähler differential module of R and
be the p-th Kahler differential forms. Given a right Poisson module M over the Poisson algebra R, there is a canonical chain complex
(1)
where for
,
is defined as:
It is easily check that
is well defined, that is
.
The complex (1) is called the Poisson complex of R with values in M, and for
its p-th homology is called the p-th Poisson homology of R with values in M, denoted by
.
Definition 4 [2] . For any
,
be the p-fold polyderivations from R to M. There is also a canonical cochain complex
(2)
where
is defined as
with
It is easily check that
is well defined, that is
.
The complex (2) is called the Poisson cochain complex of R with values in M, and for
its p-th cohomology is called the p-th Poisson cohomology of R with values in M, denoted by
.
The elements in
are called Poisson derivations, and the elements in
are called Hamiltonian derivations which are of the form
, for
.
Definition 5 [8] . A finite dimensional k-algebra R is frobenius if satisfied: as left R module,
, where
.
Definition 6 [8] . Let A be a Frobenius Poisson algebra with defining bilinear form
. Define a map
via
, for
. We call D the modular derivation
Proposition 3 [2] . Let
be a Poisson derivation, and M be a right Poisson R-module. Define a new bilinear map
as
.
Then the R-module M with
is a right Poisson R-module, which is called the twisted Poisson module of M twisted by the Poisson derivation D, denoted by
.
Lemma 1 [11] . Let S be a Frobenius Poisson algebra. Then, for all
, we have:
4. Homology Group
Before we calculate the homology, it is necessary to write the homology complex first. We give the basis in linear space in every point of the complex. Obviously,
.
,
,
and
or 1, for
.
, and
.
,
.
Similarly, we can get
So we can get the Poisson complex
(3)
Then we calculate the Poisson homology group and get some results.
Proposition 4.1.
Proof:
With the definition of Poisson bracke, we can write
as the results of image
. Next, we will prove that, when
, we can find preimage in
for all elements
1) The preimage of image with the length of 2
2) The preimage of image with the length of 3
3) The preimage of image with the length of n
4) Generally, The preimage of image with the length of
we can find preimage in
for all elements
Hence, we can find preimage of the image in
when image length
satisfied
. However, we can not find the preimage of image with the length of 1 in
.
So we can get 0-th homology group
.
The proof is completed.
Proposition 4.2.
Proof:
Firstly, we calculate
the mapping between the elements is
We have n preimage, each of them has the length of
. The image has
in length, then we calculate the image in each item.
The first term:
obviously, image has the different form with different value in
. Now we discuss all the situation about the first item in
Table 1.
The second item:
Similarly, we discuss all the situation about the second item in Table 2.
......
The
-th item:
The n-th item:
Similarly, we can discuss all the situation about the
-th, n-th item.
Because the Poisson structure of L is homogeneous. We can discuss the image of
by length.
1) The preimage of image with the length of 1
Let
, then
. we can get
for
, that is
. we have
Table 1. This table shows different form in image with the different value in
Table 2. This table shows different form in image with the different value in
Obviously, for
, the preimage of
is
under the map
.
2) The preimage of image with the length of 2
Let
, then
. we can get
for
, that is
.
we have:
Obviously, for
, the preimage of
is
under the map
.
3) Generally, the preimage of image with the length of
Let
, then
.
we have:
Obviously, under the map
, for
, the preimage of
is
.
(4)The preimage of image with the length of
Let
, then
.
we have:
Obviously, under the map
, for
, the preimage of
is
.
In conclusion,
is surjection. So we have the 1-th homology group
The proof is completed.
Now, we write the general condition in Proposition 4.3, the process of proof is similar with Proposition 4.2.
Proposition 4.3.
.
Proposition 4.4.
Proof: lastly, we calculate the n-th homology group
Firstly, we calculate
,
the image is:
We can easily see that this element can not be 0, that is
, so
The proof is completed.
In conclusion, we have the homology group of L
(4)
5. Modular Derivation and Twisted Poisson
In this part, we should calculate the Frobenius modular derivation and get the twisted Poisson structure. Firstly, we need get the Frobenius isomorphism and Frobenius pairing.
5.1. Frobenius Isomorphism and Frobenius Pairing
The Frobenius algebra L has dimension 2n, with basis
. The dual space
has the basis
, satisfied
Frobenius isomorphism
is given by
Choosing a basis
.
We can easily get the Frobenius paring is
5.2. Twisted Poisson Module
Now we calculate the Modular derivation
Compute it by each item
(5)
(6)
(7)
(8)
By the definition of
, Equations (5)-(7) equal to 0. Then we discuss (8).
when
,
We have
We can get
(9)
Then we give the twisted Poisson module of L
(10)
6. Twisted Poisson Homology
In this part, we calculate the twisted Poisson homology after we get the new poisson module structure
. give a right Poisson module
, we have the new canonical chain complex
(11)
where
,
is defined by:
Then we calculate the twisted Poisson homology group and get some results.
6.1. 0-th Twisted Poisson Homology
we will prove that we can find preimage in
for all elements when
1) The preimage of image with the length of 1
2) The preimage of image with the length of 2
3) The preimage of image with the length of 3
......
4) The preimage of image with the length of n
Hence, we can find preimage of the image
in
, the preimage is
But for the constant term k (with the length of 0), we can not find the preimage. So we can get 0-th twisted homology group
6.2. 1-th Twisted Poisson Homology
the mapping between the elements is
We have n preimage, and each of them has the length of
, and the image has
in length. Similarly, we can calculate the image in each item like in part 3, we do not repeat.
Because the Poisson structure of L is homogeneous. We can discuss the image of
by length.
1) The preimage of image with the length of 1
Let
, then
.
for
,
.
We have
We can see that the image
and
in
has the same coefficient, it is conflict with our results. So the image with length 1 in
can not find the preimage in
. In the same time, the constant term k (with the length of 0) also can not find the preimage. So we can get 1-th twisted homology group.
2) The preimage of image with the length of 2
Let
, then
for
, that is
.
We have
Obviously, for
, the preimage of this image is
under the map
.
3) Generally, the preimage of image with the length of
Let
, then
We have,
for
, the preimage of this element is
.
4) The preimage of image with the length of
Let
,
then
.
We have
for
, the preimage of this element is
In conclusion,
is not surjection. So we have the 1-th homology group
6.3. 2-th Twisted Poisson Homology
That is
we calculate the image in each item
The first term
......
The
-th term
Next, For every element in
, we try to find the preimage in
.
We just take one item for example
Similarly, because the Poisson structure of L is homogeneous. We can discuss the image of
by length.
1) The preimage of image with the length of 1
Let
, then
Then for
,
That is
.
we have
2) The preimage of image with the length of 2
Let
, then
for
,
that is
We have
We can see that the image
and must have the same coefficient, it is conflict with our results. So the image with length 1 and 2 in
can not find the preimage in
. In the same time, the constant term k (with the length of 0) also can not find the preimage. Now we discuss the elements in
when the length ≥ 3, we will prove that we can get the preimage in
3) The preimage of image with the length of 3
Let
, then
We have
Obviously, for
, the preimage of this image is
under the
.
4) The preimage of image with the length of
Let
, then
We have
Obviously, the preimage of this image is
under the
.
In conclusion,
is not surjection. So we have the 2-th twisted homology group
In general situation, for
, the m-th twisted homology
has the elements in
with length of
.
6.4. n-th Twisted Poisson Homology
We calculate the Poisson bracket
We can easily see that this element can not be 0, that is
, so
7. Twisted Poincaré Duality between Poisson Homology and Cohomology
In this part, we will check the Twisted Poincaré duality between Poisson homology and cohomology, that is for
, we have
Next, we need to calculate the cohomology group.
obviously, for all
, the equation established
(12)
For every
, p-fold polyderivations from L to L, denoted
,
. recall the canonical cochain complex
For every
and
.
is defined by
obviously,
and
.
Let
, and assume it satisfies
Then Equation (12) lead successively to
For
and
, we have
.
Hence, we can get the Proposition 7.1.
Proposition 7.1.
.
Proof: for every
, d is uniquely determined by the values of
,
,
,
.
Moreover, d must satisfy the relations
, since d is a derivation, so we have
.
So we have the that for
,
. Hence the space
has basis
, where
1) for
,
, the derivation
is defined by
2) for
,
, the derivation
is defined by
......
(n) for
,
, the derivation
is defined by
In particular,
. Let
if and only if d satisfied:
(13)
In Equation (13), we have:
From the coefficient of
we can get for
, we have
The proof is completed.
Proposition 7.2.
.
Proof: let
.
Set
, from Equation (12) we can get
is a Poisson derivation and satisfied
. From Equation (13) we can get that for all
, we have
. That is
. Similarly,
Let
.
Let a new set
.
, so we have
.
That is the image of
in
.
The proof is completed.
Similarly, we can calculate the m-th cohomology
.
From the propostion 7.1 and propostion 7.2 we can easily get that the demension of m-th cohomology group
equal to the demension of m-th Twisted Poisson homology,
, we check the twisted Poincaréduality between them.
8. Conclusion
In this paper, we successfully calculate the i-th homology group of the algebra
in part 4. In part 5, after getting Frobenius pairing, we calculate the modular derivation and then have the twisted Poisson module structure. Furthermore, we calculate the twisted Poisson homology group in part 6. Lastly, we verified the twisted Poincaré duality between Poisson homology and Poisson Cohomology through the cohomology group of L in part 7. In future studies, we will discuss whether all these conclusions hold up for general algebra.
Cite this paper
Wang, M.Y. (2018) Poisson (Co)homology of a Class of Frobenius Poisson Algbras. Journal of Applied Mathematics and Physics, 6, 530-553. https://doi.org/10.4236/jamp.2018.63048
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