Associative superalgebras with homogeneous symmetric structures

TL;DR

This paper classifies associative superalgebras with homogeneous symmetric structures, showing restrictions on their types, characterizing simple cases, and introducing generalized double extensions for their construction.

Contribution

It provides a complete classification of associative superalgebras with homogeneous symmetric structures, including simple cases and new inductive construction methods.

Findings

Simple associative superalgebras admit either even or odd symmetric structures.

Associative superalgebras with non-zero product cannot have both even and odd symmetric structures.

Introduction of generalized double extensions for constructing and describing these superalgebras.

Abstract

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e. even or odd) and associative bilinear form on A. In this paper, we show that any associative superalgebra with non null product can not admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.