On Hom-Lie Pseudo-Superalgebras

Shengxiang Wang,Tingting Tao

doi:10.4236/apm.2016.66029

Shengxiang Wang, Tingting Tao

Open Access

https://doi.org/10.4236/apm.2016.66029

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Abstract

The aim of this article is to introduce the notion of Hom-Lie H-pseudo-superalgebras for any Hopf algebra H. This class of algebras is a natural generalization of the Hom-Lie pseudo-algebras as well as a special case of the Hom-Lie superalgebras. We present some construction theorems of Hom-Lie H-pseudo-superalgebras, reformulate the equivalent definition of Hom-Lie H-pseudo-super-algebras, and consider the cohomology theory of Hom-Lie H-pseudo-superalgebras with coefficients in arbitrary Hom-modules as a generalization of Kac’s result.

Highlights

The notion of conformal algebras [1]-[5] was introduced by Kac as a formal language describing the singular part of the operator product expansion in two-dimensional conformal field theory, and it came to be useful for investigation of vertex algebras
The concept of vertex algebras was derived from mathematical physics; it was first mathematically defined and considered by Borcherds in [9] to obtain his solution of the Moonshine conjecture in the theory of finite simple groups
Classification problems, cohomology theory and representation theory have been considered in [10]-[12]

Summary

Introduction

The notion of conformal algebras [1]-[5] was introduced by Kac as a formal language describing the singular part of the operator product expansion in two-dimensional conformal field theory, and it came to be useful for investigation of vertex algebras (see [6]-[8]). In [24], Hom-algebras and Hom-coalgebras were introduced by Makhlouf and Silvestrov as a generalization of ordinary algebras and coalgebras in the following sense: the associativity of the multiplication was replaced by the Hom-associativity and similar for Hom-coassociativity They defined the structures of Hom-bialgebras and Hom-Hopf algebras, and described some of their properties extending properties of ordinary bialgebras and Hopf algebras in [25] and [26]. We think whether we can extend the notion of cohomology groups to Hom-Lie H-pseudo-superalgebras. This becomes our second motivation of the paper.

Preliminaries

Hom-Pseudo Superalgebras of Associative and Lie Types

Hom-Annihilation Superalgebras

Hom-Lie Conformal Superalgebras

Cohomology of Hom-Lie H-Pseudo-Superalgebras

Choosing a splitting