Rigid Analytic Quantum Groups

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Dupré, Nicolas

Advisor dc:contributor.advisor

• Wadsley, Simon James

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

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b266428f-8711-4386-b9ca-6b4ff4a3f4c5/download
• https://creativecommons.org/licenses/by-nc-sa/4.0/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.38204

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/291025