Nilpotentizers and the Nilpotent Graphs: Structural Insights into Lie Superalgebras

TL;DR

This paper explores the structure of Lie superalgebras through the concepts of nilpotentizer and nilpotent graph, revealing invariants and introducing a new measure of nilpotency using category theory.

Contribution

It introduces the nilpotentizer and nilpotent graph as new tools for analyzing Lie superalgebras and establishes their fundamental properties and invariance.

Findings

Nilpotent graph is an isomorphic invariant of Lie superalgebras.

Introduces a nilpotency measure for Lie superalgebras.

Connects Lie superalgebras with their nilpotent substructures via category theory.

Abstract

In this paper, we systematically investigate the nilpotentizer and nilpotent graph for a Lie superalgebra over the field of characteristic not equal to 2. First, we establish some fundamental properties of the nilpotentizer. Next, we show that the nilpotent graph is one of the isomorphic invariants of Lie superalgebras. Furthermore, we introduce the nilpotency measure which provides a quantitative assessment of nilpotency for a Lie superalgebra. Finally, we use category theory to establish connections between Lie super?algebras and their nilpotent substructures, based on the construction of the nilpotentizer.