A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

TL;DR

This paper develops a categorification framework for finite-dimensional irreducible representations of quantum sl(2) and their tensor products, using Harish-Chandra bimodules and geometric methods, linking algebraic and geometric categorifications.

Contribution

It introduces new categorifications of quantum sl(2) representations via Harish-Chandra bimodules and geometric models, connecting canonical bases with bimodule categories.

Findings

Categorification via Harish-Chandra bimodules for gl(n)

Geometric categorification using flag varieties

Categorical Schur-Weyl duality and basis interpretations

Abstract

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gl(n). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed. We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.