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Crystal Methods in Representation Theory
Student-Run Research| Speaker: | Philip Sternberg, UC Davis |
| Location: | 693 Kerr |
| Start time: | Wed, Jun 8 2005, 12:10PM |
Description
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.