# Mathematics Colloquia and Seminars

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### Tensor factorization and Spin construction for affine Lie algebras

**Algebra & Discrete Mathematics**

Speaker: | Rajeev Walia, Michigan State U. |

Location: | 1147 MSB |

Start time: | Thu, Jan 25 2007, 3:10PM |

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.