Quantum group actions on rings and equivariant K-theory

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.