Maximal subalgebras of the general linear Lie algebra containing Cartan subalgebras

Abstract

Let Γ be a finite group and V a finite-dimensional Γ-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 gl(V, Γ). In the situation of Γ being a cyclic group, we determine the minimal dimensions of pre-nil or nil faithful representations for any finite-dimensional abelian Lie color algebra.