Abstract

Quantum groups are deformations of classical algebras such as enveloping algebras of Lie algebras or coordinate rings of algebraic groups. A common theme about these objects is that they should still share similar properties as their classical counterparts, and so for instance in the right context we can think of quantum groups as algebras of “polynomial” functions on some “quantum space”. When working over p-adic fields, there is a very natural notion of (locally) analytic functions and one can define a so-called rigid analytic space, which is essentially a space whose functions are locally analytic. Therefore one can ask whether there is a good notion of algebras of analytic functions on a “quantum p-adic space”, or said differently a notion of rigid analytic quantum group. We will show how to construct an analytic version of the quantized enveloping algebra of sl_2, and we will then describe some of the properties that this algebra satisfies which make its representation theory possible to study.