The wonderful compactification for quantum groups

This thesis studies the asymptotics of quantum groups using an approach centered on the wonderful compactification. The wonderful compactification of a semisimple group was introduced by De Concini and Procesi, and has become an important tool in geometric representation theory. We provide an exposition of several constructions of the wonderful compactification in order to illustrate how it links the geometry of the group to the geometry of its partial flag varieties, and how it encodes the asymptotics of matrix coefficients for the group. We then construct quantum group versions of the wonderful compactification, its associated Vinberg semigroup, its stratification by G × G orbits, and its algebra of differential operators. A key technical aspect of our approach is the notion of a noncommutative projective scheme associated to a ring graded by a lattice. We provide explicit descriptions of our constructions in the case of SL 2, explain connections to previous work on the flag variety of a quantum group, and discuss conjectural applications of the newly-defined objects that appear in this thesis.