Graded simple Lie algebras and graded simple representations

Abstract

Let Q be an abelian group and k a field. We prove that any Q-graded simple Lie algebra g over k is isomorphic to a loop algebra in case k has a primitive root of unity of order |Q|, if Q is finite, or k is algebraically closed and dim g< | k| (as cardinals). For Q-graded simple modules over any Q-graded Lie algebra g, we propose a similar construction of all Q-graded simple modules over any Q-graded Lie algebra over k starting from nonextendable gradings of simple g-modules. We prove that any Q-graded simple module over g is isomorphic to a loop module in case k has a primitive root of unity of order |Q| if Q is finite, or k is algebraically closed and dim g< | k| as above. The isomorphism problem for simple graded modules constructed in this way remains open. For finite-dimensional Q-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.