Z-graded Hom-Lie Superalgebras

TL;DR

This paper introduces Z-graded hom-Lie superalgebras, explores their maximal and minimal structures, and studies invariant bilinear forms and simplicity conditions within this algebraic framework.

Contribution

It defines Z-graded hom-Lie superalgebras, constructs maximal and minimal examples, and extends invariant bilinear forms from local parts to entire structures, also analyzing simplicity conditions.

Findings

Existence of maximal and minimal Z-graded hom-Lie superalgebras for given local structures.

Extension of invariant bilinear forms from local to entire superalgebras.

Conditions under which Z-graded hom-Lie superalgebras are simple.

Abstract

In this paper we introduce the notions of Z-graded hom-Lie superalgebras and we show that there is a maximal (resp., minimal) Z-graded hom-Lie superalgebra for a given local hom-Lie superalgebra. Morever, we introduce the invariant bilinear forms on a Z-graded hom-Lie superalgebra and we prove that a consistent supersymmetric {\alpha}-invariant form on the local part can be extended uniquely to a bilinear form with the same property on the whole Z-graded hom-Lie superalgebra. Furthermore, we check the condition in which the Z-graded hom-Lie superalgebra is simple.