Tensor factorization and Spin construction for Kac–Moody algebras

Abstract

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac–Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac–Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras. We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac–Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g -representations for which, surprisingly, the Spin functor gives a g -representation in Bernstein–Gelfand–Gelfand category O . Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.