Mathematics Colloquia and Seminars

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Given a module $V$ over a Kac-Moody Lie algebra $\mathfrak{g}$, we can produce a "quantum deformation" of this picture parameterized by a complex number $q$. This is called a deformation because when $q=1$, the classical algebra and module are recovered.

When we set $q=0$, we are left with a discrete version of our original picture that retains almost all of the information we started with; In particular, weight multiplicities and tensor product multiplicities are preserved. We will see how "crystallization" gives purely combinatorial methods for solving classical representation theory problems, as well as motivating and answering new questions about the representation theory of Lie algebras.