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CRYSTALLIZING FUNCTIONS ON COMPACT LIE GROUPS

Speaker(s)
ROBERT YUNCKEN
Affiliation
Université Clermont Auvergne, France
Date
June 7, 2023, 5:15 p.m.
Information about the event
405 IMPAN & ZOOM
Seminar
North Atlantic Noncommutative Geometry Seminar

The theory of crystal bases, due to Kashiwara and Lusztig, is a means of simplifying the representation theory of semisimple Lie algebras by passing through quantum groups. Specifically, varying the parameter q of a quantized enveloping algebra, we pass from the classical theory at q=1 through the Drinfeld-Jimbo algebras at 0 < q < 1 to the crystal limit at q=0.  At this point, the main features of the representation theory (matrix coefficients, Clebsch-Gordan coefficients, branching rules) crystallize into purely combinatorial data described by crystal graphs. In this talk, we will describe what happens to the *-algebra of functions on a compact semisimple Lie group under the crystallization process, leading to a higher-rank graph algebra. This generalizes, in part, work by Woronowicz, Hong and  Szymański, and Giselsson. (Joint work with Marco Matassa.)