Mathematics Colloquia and Seminars

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We will discuss the “Factorization Phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into smaller representations of the subalgebra. The original Lie algebra may be finite-dimensional and semi-simple, or an affine Kac-Moody algebra. We will provide an algebraic explanation for such a phenomenon using “Spin construction”. First we will give a few Factorization results in finite-dimension for any embeding of a Lie algebra into another, using Spin construction and give some combinatorial consequences of it. In order to generalize these results, we will extend the notion of Spin to affine Lie algebras which requires a very delicate treatment. We will introduce a class of “Orthogonal Level zero” representations for which, surprisingly, the Spin gives a class of representations which always have a non-zero Level. Next, we will give the formula for the character of Spin of an orthogonal representation and a few factorization results for affine Lie algebras. Finally, we discuss classification of “Coprimary representations” i.e those representations whose Spin is irreducible.