## Rigid Analytic Quantum Groups

dc.contributor.author | Dupré, Nicolas | |

dc.date.accessioned | 2019-04-01T14:46:03Z | |

dc.date.available | 2019-04-01T14:46:03Z | |

dc.date.issued | 2019-04-27 | |

dc.date.submitted | 2018-09-28 | |

dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/291025 | |

dc.description.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. | |

dc.language.iso | en | |

dc.rights | Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) | |

dc.rights.uri | https://creativecommons.org/licenses/by-nc-sa/4.0/ | |

dc.subject | Quantum groups | |

dc.subject | D-modules | |

dc.subject | p-adic representation theory | |

dc.title | Rigid Analytic Quantum Groups | |

dc.type | Thesis | |

dc.type.qualificationlevel | Doctoral | |

dc.type.qualificationname | Doctor of Philosophy (PhD) | |

dc.publisher.institution | University of Cambridge | |

dc.publisher.department | Pure Mathematics and Mathematical Statistics | |

dc.date.updated | 2019-04-01T13:46:17Z | |

dc.identifier.doi | 10.17863/CAM.38204 | |

dc.publisher.college | Girton | |

dc.type.qualificationtitle | PhD in Mathematics | |

cam.supervisor | Wadsley, Simon James | |

cam.thesis.funding | true |