Quantum group actions on rings and equivariant K-theory

TL;DR

This paper develops an equivariant K-theory framework for quantum spaces with quantum group symmetry, enabling the computation of K-groups for quantum homogeneous spaces and symmetric algebras.

Contribution

It introduces a new equivariant K-theory for quantum group actions on noncommutative spaces using Quillen's exact categories, with explicit computations for key examples.

Findings

Computed equivariant K-groups for quantum homogeneous spaces.

Derived K-theory for quantum symmetric algebras of classical type.

Established a method for calculating equivariant K-theory in noncommutative geometry.

Abstract

Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an equivariant K-theory of such quantum vector bundles using Quillen's exact categories, and provide means for its compution. The equivariant K-groups of quantum homogeneous spaces and quantum symmetric algebras of classical type are computed.