A category O for quantum Arens-Michael envelopes.

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• Nicolas Dupré
• Friday 12 October 2018, 15:00-16:00
• CMS, MR14.

If you have a question about this talk, please contact Richard Freeland.

Classically, the BGG category O of a complex semisimple Lie algebra is a subcategory of its category of representations which is particularly well-behaved. It contains all the highest weight modules and so in particular all the finite dimensional representations, and it has nice combinatorics (e.g. BGG reciprocity). There is a natural analogue of this category for quantum groups, which more precisely corresponds to the integral category O (i.e. the direct sum of all the integral blocks). A few years ago, Tobias Schmidt defined a category O for the Arens-Michael envelope of the enveloping algebra of a p-adic Lie algebra. This Arens-Michael envelope can be defined as a certain Fréchet completion of the enveloping algebra, and it satisfies certain properties which makes it what is called a Fréchet-Stein algebra. These algebras have a nice category of modules, called coadmissible, and Schmidt defined his category O as a certain subcategory of the category of all coadmissible modules. His main result was that his category is equivalent to the usual category O of the Lie algebra. In this talk, we will explain how to construct a quantum analogue of the Arens-Michael envelope and a category O for it. We will then see that the analogue of Schmidt’s theorem is also true for our category.

This talk is part of the Junior Algebra and Number Theory seminar series.

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