On the maximal abelian subalgebras of the general linear Lie color algebras

TL;DR

This paper classifies maximal abelian subalgebras of general linear Lie color algebras over algebraically closed fields, focusing on their structure and minimal faithful representations when the grading group is cyclic.

Contribution

It provides a classification of pre-nil and nil maximal abelian subalgebras of Lie color algebras and determines minimal dimensions of faithful representations for cyclic group gradings.

Findings

Classification of maximal abelian subalgebras in Lie color algebras

Determination of minimal dimensions of faithful representations for cyclic gradings

Explicit descriptions in the case of cyclic grading groups

Abstract

Let $\Gamma$ be a finite group and $V$ a finite-dimensional $\Gamma$-graded space over an algebraically closed field of characteristic not equal to 2. In the sense of conjugation, we classify all the so-called pre-nil or nil maximal abelian subalgebras for the general linear Lie color algebra $\frak{gl}(V,\Gamma)$. In the situation of $\Gamma$ being a cyclic group, we determine the minimal dimensions of pre-nil or nil faithful representations for any finite-dimensional abelian Lie color algebra.