## Rigid Analytic Quantum Groups

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##### Authors

Dupré, Nicolas

##### Advisors

Wadsley, Simon James

##### Date

2019-04-27##### Awarding Institution

University of Cambridge

##### Author Affiliation

Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Dupré, N. (2019). Rigid Analytic Quantum Groups (Doctoral thesis). https://doi.org/10.17863/CAM.38204

##### Abstract

Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\wideparen{U_q(\mathfrak{g})}$ and $\wideparen{\mathcal{O}_q(G)}$ of the quantized enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(\mathfrak{g})$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $\mathcal{O}$ for $\wideparen{U_q(\mathfrak{g})}$.
We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of $\wideparen{\mathcal{O}_q(G)}$. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object $\widehat{\D_q^\lambda}$ which plays the role of the sheaf of $\lambda$-twisted differential operators. We show that when $\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of $\widehat{\mathcal{D}_q^\lambda}$.
The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig's integral form of the quantum Borel.

##### Keywords

Quantum groups, D-modules, p-adic representation theory

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.38204

##### Rights

Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)

Licence URL: https://creativecommons.org/licenses/by-nc-sa/4.0/