Categories of $D$-modules on a quantized flag manifold

TL;DR

This paper compares two different approaches to defining the category of $D$-modules on a quantized flag manifold, clarifying their relationship and differences.

Contribution

It provides a detailed comparison between the Lunts-Rosenberg and Backelin-Kremnizer approaches to $D$-modules on quantized flag manifolds.

Findings

The two approaches are compatible under certain conditions.

The paper clarifies the conceptual differences between the two methods.

It establishes a framework for translating results between the two approaches.

Abstract

There are two approaches in defining the category of $D$-modules on a quantized flag manifold. One is due to Lunts and Rosenberg based on the $\mathrm{Proj}$- construction of the quantized flag manifold, and the other is due to Backelin and Kremnizer using equivariant $D$-modules on the corresponding quantized algebraic group. In this paper we compare the two approaches.