Graded simple Lie algebras and graded simple representations

摘要

For any finitely generated abelian group $Q$, we reduce the problem ofclassification of $Q$-graded simple Lie algebras over an algebraically closedfield of "good" characteristic to the problem of classification of gradings onsimple Lie algebras. In particular, we obtain the full classification offinite-dimensional $Q$-graded simple Lie algebras over any algebraically closedfield of characteristic $0$ based on the recent classification of gradings onfinite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-gradedLie algebra (not necessarily simple) to classification of gradings on simplemodules. For finite-dimensional $Q$-graded semisimple algebras we obtain agraded analogue of the Weyl Theorem.