Rigid analytic quantum groups

Abstract

Following constructions in rigid analytic geometry, we introduce a theory of p-adic analytic quantum groups. We first define Fréchet completions U̇q(g) and ̊ Oq(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 U̇q(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 ̊ Oq(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 ” Dλ q 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 ” Dλ q . The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion ◊ Oq(B) of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over ◊ Oq(B) can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig’s integral form of the quantum Borel.