The wonderful compactification for quantum groups

TL;DR

This paper develops a quantum analogue of the wonderful compactification of algebraic groups, using noncommutative geometry and a quantum Vinberg semigroup, extending classical geometric concepts to the quantum setting.

Contribution

It introduces a quantum version of the wonderful compactification via a noncommutative projective scheme and defines a quantum Vinberg semigroup as a q-deformation of a Rees algebra.

Findings

Constructed a quantum wonderful compactification as a noncommutative projective scheme.

Defined a quantum Vinberg semigroup compatible with Poisson structures.

Provided explicit computations for the case of SL_2.

Abstract

In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of $\mathsf{Proj}$ categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a $q$-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of $\mathrm{SL}_2$.