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Title: Rigid analytic quantum groups
Author: Dupré, Nicolas
ISNI:       0000 0004 7961 8718
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2019
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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.
Supervisor: Wadsley, Simon James Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: Quantum groups ; D-modules ; p-adic representation theory