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

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 $\frak{gl}(V,Γ)$. 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.