Factorizable sheaves and quantum groups

TL;DR

This paper geometrically constructs the category of representations of Lusztig's small quantum group at a root of unity, revealing its modular structure through configuration spaces.

Contribution

It introduces a geometric approach to understanding the representation category of small quantum groups, linking algebraic and geometric perspectives.

Findings

Categorical structure of small quantum group representations is described geometrically.

Modular properties of the category are elucidated via configuration spaces.

Provides new tools for studying quantum groups through geometry.

Abstract

The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.