Quantum McKay Correspondence and Equivariant Sheaves on the Quantum Projective Line

TL;DR

This paper constructs a derived category of G-equivariant sheaves on the quantum projective line at roots of unity using quantum McKay correspondence, linking it to quiver representations and root systems.

Contribution

It introduces a new derived category framework for G-equivariant sheaves on the quantum projective line, connecting quantum McKay correspondence with quiver and root lattice categorification.

Findings

Constructed a dg-algebra analogue of the structure sheaf at roots of unity.

Related the derived category to quiver representations of A,D,E types.

Categorified the root lattice and identified indecomposable sheaves with root system elements.

Abstract

In this paper, using the quantum McKay correspondence, we construct the "derived category" of G-equivariant sheaves on the quantum projective line at a root of unity. More precisely, we use the representation theory of U_{q}sl(2) at root of unity to construct an analogue of the symmetric algebra and the structure sheaf. The analogue of the structure sheaf is, in fact, a complex, and moreover it is a dg-algebra. Our derived category arises via a triangulated category of G-equivariant dg-modules for this dg-algebra. We then relate this to representations of the quiver (\Gamma, \Om), where \Gamma is the A,D,E graph associated to G via the quantum McKay correspondence, and \Om is an orientation of \Gamma. As a corollary, our category categorifies the corresponding root lattice, and the indecomposable sheaves give the corresponding root system.