Twisted Poisson Homology of Truncated Polynomial Algebras in Four Variables

In this paper, we study the twisted Poisson homology of truncated polynomials algebra A in four variables, and we calculate exactly the dimension of i-th (i = 1, 2, 3, 4) twisted Poisson homology group over A by the induction on the length. The calculation methods provided in this paper can also solve truncated polynomials algebra in a few variables.


Introduction
For a Poisson algebra, Lichnerowicsz (see [1]) first introduced the notion of Poisson cohomology in 1977.This Poisson cohomology provides important information about the structure of Poisson algebra.Launois S and Richard L (see [2]) studied the Poisson (co)homology of the algebra of the truncated polynomial in two variables and established a duality between the two.Can Zhu (see [3]) proved that this result is still true for all Frobenius Poisson algebra as follows (see [3], Corollary 3.3).
In general, given a Poisson algebra, it is very difficult to calculate its Poisson cohomology.From the above Theorem, the dimension of Poisson cohomology space is determined by calculating twisted Poisson homology.So there is a natu-ral problem: how to calculate the twisted Poisson homology of a Poisson algebra.
For example, for algebra in [2], how would we calculate its twisted Poisson homology if we extended two variables to four variables or even n variables.The purpose of this paper is to provide a solution to calculate the twisted Poisson homology of truncated polynomials algebra in four variables.
In this article, we will recall some basic knowledge in the second part and show the main conclusions in the third part.

Preliminaries
Throughout,  is a field of characteristic zero.Definition 1 [4].A right Poisson module M over the Poisson algebra R is a k-vector space M endowed with two bilinear maps ⋅ and { } Left Poisson modules are defined similar.Any Poisson algebra R is naturally a right or left Poisson module over itself.
Definition 2 [5].Let A be a Poisson algebra.In general, let ( ) 1 A Ω be the Kähler differential module of A and ( ) ( ) be the p-th Kähler differentia forms, where ∧ is the wedge product over A (also in [6]).Given a right Poisson module M over the Poisson algrbra A, there is a canonical chain complex ( ) ( ) ( ) ( ) where for 1 p ≥ , ( ) ( )

Twisted Poisson Homology of Truncated Polynomial
We consider the truncated polynomials algebra [ ] , , , , is clear from the definition of Poisson bracket.We can get the modular derivation ( ) ( ) ( ) ( ) Then we define a new bilinear map { } By Now we can get some conclusions as follows.
( ) ( ) First of all, we have that ( ) So we just need to consider which elements in ( ) have the inverse image.
We proceed by the induction on the length of the elements in ( ) Remark: We make an agreement on the length: the length of 1) The image of element of length 0 { } ( ) ( ) , , , , , , , , Obviously, we get: 3) The image of element of length 2
( ) ( ) ( ) In this part, we need to consider two questions: 1) what is the form of the element in 1 ker π δ ; 2) whether the element in 1 ker π δ has the inverse image.
We distinguish four cases below.
1) The element with the length of 0 Since ( ) { } ( ) and all elements with the form as 1 i dx ⊗ have the same situation.
2) The element with the length of 1 Now we prove that these elements with the length of 1 in 3) The element with the length of 2 ( ) ( ) We can find the inverse image of all elements as the above by following The element with the length of 3 For ( ) Since the map p π δ keeps the variable unchanged, so we can't find the inverse image of i j k t x x x dx ⊗ under the map 2 π δ .
In conclusion, only ( ) Similar to the proof of proposition 3.2, we can prove the following Proposition 3.3. Proof.
( ) ( ) ( ) 2) The element with the length of 1 Obviously, the preimage of , D HP A A .It is clear that the element with the same form have the same situ- ation.
3) The element with the length of 2   Proof.

(
Theorem 1): Theorem 1.Let S be a Frobenius Poisson algebra.Then we have the following isomorphism: i N ∈ , where S σ is the Poisson module induced by the Frobenius isomorphism : S S σ * → Poisson derivation, and M be a right Poisson A-module.Define a new bilinear map { } Then the A-module with { } , D M − − is a right Poisson A-module, which is called the twisted Poisson module of M twisted by the Poisson derivation D, denote by D M .
definition 3, D A becomes a twisted Poisson right A-module with { } , result and definition 2, we obtain a new canonical chain complex over A:

1 )
The element with the length of 0 When we calculate the 1-th twisted Poisson homology group, we have found that each element of length 0 in When we calculate the 2-th twisted Poisson homology group, we have noticed that each element like 1 .