An application of Lie superalgebras to affine Lie algebras

AbstractLet g be a finite dimensional complex simple Lie algebra and ĝ the associated affine Lie algebra. Let V be a finite dimensional irreducible g-module and X an integrable highest weight ĝ-module. We show that the tensor product of X with the space L(V) of loops in V is reducible if the highest weight of X is “large” compared with that of V. (This complements the result of a previous paper in which we showed that the tensor product is irreducible if the highest weight of X is “small” compared with that of V.) The proof makes use of a Lie superalgebra constructed from ĝ and V