  The paper extends existing Lie algebra representation theory related to Lie
algebra gradings. The notion of a representation compatible with a given
grading is defined and applied to finite-dimensional representations of
$sl(n,\mathbb{C})$ in relation with its $\mathbb{Z}_2$-gradings. For
representation theory of $sl(n,\mathbb{C})$ the Gel'fand-Tseitlin method turned
out very effective.
