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 $Ø$. 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 classification of those representations for which $\Spin$ is irreducible.