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Nilpotentizers and the Nilpotent Graphs: Structural Insights into Lie Superalgebras

Baojin Zhang, Liming Tang

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.

Nilpotentizers and the Nilpotent Graphs: Structural Insights into Lie Superalgebras

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.
Paper Structure (5 sections, 16 theorems, 68 equations)

This paper contains 5 sections, 16 theorems, 68 equations.

Table of Contents

  1. Basic
  2. The nilpotentizer of a Lie superalgebra
  3. The nilpotent graph of a Lie superalgebra
  4. The category perspective of the nilpotentizer
  5. Acknowledgement

Key Result

Proposition 3

Let $L=L_{\overline{0}}\oplus L_{\overline{1}}$ be a finite dimensional Lie superalgebra, and let $J$ be a graded ideal of $L$ and $x,z\in L$. Then

Theorems & Definitions (42)

  • Definition 1
  • Example 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • ...and 32 more