An update on Heisenberg and Kac-Moody categorification

TL;DR

This paper advances the understanding of Heisenberg and Kac-Moody categorification by providing explicit formulas for complex adjunctions, simplifying the theoretical framework in type A representation theory.

Contribution

It introduces explicit formulas for the challenging adjunction in Kac-Moody categorification derived from Heisenberg actions, enhancing the existing theory.

Findings

Explicit formulae for the difficult adjunction are derived.

Simplifications to the existing categorification theory are achieved.

The work deepens understanding of the relationship between Heisenberg and Kac-Moody actions.

Abstract

Heisenberg categories act on many Abelian categories appearing in type A representation theory. There is also a general procedure to construct from a Heisenberg action another action of a Kac-Moody 2-category for some associated Cartan matrix. One of the adjunctions on the Kac-Moody side is matched up in an easy way with adjunctions on the Heisenberg side, but the second adjunction is much harder to describe. In this paper, we derive explicit formulae for this difficult adjunction, leading to some further simplifications to the existing theory.