Localization of quantum biequivariant D-modules and q-W algebras

TL;DR

This paper extends localization theorems for quantum D-modules to biequivariant settings and establishes an equivalence between categories of equivariant quantum group modules and q-W algebra modules, generalizing Skryabin's result.

Contribution

It introduces a biequivariant localization theorem for quantum D-modules and links equivariant quantum group modules to q-W algebra modules, expanding the theoretical framework.

Findings

Established biequivariant localization theorem for quantum D-modules.

Proved an equivalence between equivariant quantum group modules and q-W algebra modules.

Extended Skryabin's equivalence to an equivariant quantum setting.

Abstract

We present a biequivariant version of Kremnizer-Tanisaki localization theorem for quantum D-modules. We also obtain an equivalence between a category of finitely generated equivariant modules over a quantum group and a category of finitely generated modules over a q-W algebra defined in arXiv:1011.2431. This equivalence can be regarded as an equivariant quantum group version of Skryabin equivalence. The biequivariant localization theorem for quantum D-modules together with the equivariant quantum group version of Skryabin equivalence yield an equivalence between a certain category of quantum biequivariant D-modules and a category of finitely generated modules over a q-W algebra.