Structures and Low Dimensional Classifications of Hom-Poisson Superalgebras

Qingcheng Zhang,Chunyue Wang,Zhu Wei

doi:10.1155/2015/354341

Abstract

Hom-Poisson superalgebras can be considered as a deformation of Poisson superalgebras. We prove that Hom-Poisson superalgebras are closed under tensor products. Moreover, we show that Hom-Poisson superalgebras can be described using only the twisting map and one binary operation. Finally, all algebra endomorphisms on 2-dimensional complex Poisson superalgebras are computed, and their associated Hom-Poisson superalgebras are described explicitly.

Highlights

Poisson algebras are used in many fields in mathematics and physics
Poisson algebras play a fundamental role in Poisson geometry [1], quantum groups [2], and deformation of commutative associative algebras
Poisson algebras are a major part of deformation quantization, Hamiltonian mechanics [3], and topological field theories [4]

Summary

Introduction

Poisson algebras are used in many fields in mathematics and physics. Poisson superalgebras can be seen as a direct generalization of Poisson algebras. A twisted generalization of noncommutative Poisson algebras, called Hom-noncommutative Poisson algebras, are studied in [8]. In a noncommutative Hom-Poisson algebra, there exists a twisted map, a Hom-Lie bracket and a Hom-associative product. The purpose of this paper is to introduce and study a twisted generalization of Poisson superalgebras, called HomPoisson superalgebras.

Hom-Lie and Hom-Associative Superalgebras

Hom-Poisson Superalgebras

Admissible Hom-Poisson Superalgebras

A Classification of 2-Dimensional Hom-Poisson Superalgebras

If a10