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oai:www.repository.cam.ac.uk:1810/2910252019-04-02T01:26:54Zcom_1810_213747com_1810_256064col_1810_213748

Rigid Analytic Quantum Groups 10.17863/CAM.38204 Dupré, Nicolas Wadsley, Simon James 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. University of Cambridge 2019-04-27 Thesis Doctoral Doctor of Philosophy (PhD) en https://www.repository.cam.ac.uk/handle/1810/291025 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/48e09fef-b4fb-4e23-acb0-5af4b40eefbb/download 6e34f3456e5d5dd2ae0f3c865c32a9d2 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b266428f-8711-4386-b9ca-6b4ff4a3f4c5/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by-nc-sa/4.0/ Quantum groups D-modules p-adic representation theory