A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors

TL;DR

This paper establishes a categorification linking the Grothendieck groups of certain blocks in category O for sl(n) with tensor powers of the fundamental sl(2)-module, using projective and Zuckerman functors to realize algebra actions.

Contribution

It provides a novel categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) through functor actions on category O blocks.

Findings

Grothendieck group identified with tensor powers of fundamental sl(2)-module

Projective functors correspond to Lusztig canonical basis elements

Zuckerman functors generate the U(sl(2)) action

Abstract

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the wall functors act as generators of the Temperley-Lieb algebra while Zuckerman functors act as generators of U(sl(2)).