On representations of Lie algebras compatible with a grading

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.