Graded simple Lie algebras and graded simple representations

Abstract

Let Q be an abelian group and Bbbk a field. We prove that any Q-graded simple Lie algebra mathfrak {g} over Bbbk is isomorphic to a loop algebra in case Bbbk has a primitive root of unity of order |Q|, if Q is finite, or Bbbk is algebraically closed and dim mathfrak {g}<|Bbbk | (as cardinals). For Q-graded simple modules over any Q-graded Lie algebra mathfrak {g}, we propose a similar construction of all Q-graded simple modules over any Q-graded Lie algebra over Bbbk starting from nonextendable gradings of simple mathfrak {g}-modules. We prove that any Q-graded simple module over mathfrak {g} is isomorphic to a loop module in case Bbbk has a primitive root of unity of order |Q| if Q is finite, or Bbbk is algebraically closed and dim mathfrak {g}<|Bbbk | 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.