  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.

Representation Theory