Graded Simple Lie Algebras and Graded Simple Representations

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristic 0 based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any Q-graded Lie algebra (not necessarily simple) to classification of gradings on simple modules. For finite-dimensional Q-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.