Rigid Analytic Quantum Groups

Change log
Dupré, Nicolas 

Following constructions in rigid analytic geometry, we introduce a theory of p-adic analytic quantum groups. We first define Fréchet completions \wideparenUq(g) and \wideparenOq(G) of the quantized enveloping algebra of a semisimple Lie algebra 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(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 O for \wideparenUq(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 \wideparenOq(G). Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of λ-twisted D-modules on this analytic quantum flag variety. This category has a distinguished object \Dqλ^ which plays the role of the sheaf of λ-twisted differential operators. We show that when λ is regular and dominant, the global section functor gives an equivalence of categories between the coherent λ-twisted D-modules and the finitely presented modules over the global sections of Dqλ^.

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.

Wadsley, Simon James
Quantum groups, D-modules, p-adic representation theory
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge