Evaluation 2-functors for Kac–Moody 2-categories of type $A_2$

Marco Mackaay, James Macpherson and Pedro Vaz

Vol. 341 (2026), No. 1, 103–145

DOI: 10.2140/pjm.2026.341.103

Abstract

We construct a 2-functor from the Kac–Moody 2-category for the extended quantum affine ${\mathfrak{𝔰}\mathfrak{𝔩}}_3$ to the homotopy 2-category of bounded chain complexes with values in the Kac–Moody 2-category for quantum ${\mathfrak{𝔤}\mathfrak{𝔩}}_3$, categorifying the evaluation map between the corresponding quantum Kac–Moody algebras. Our approach establishes and exploits a categorical analogue of the well-known relation between the evaluation map and Lusztig’s internal braid group action for quantum ${\mathfrak{𝔤}\mathfrak{𝔩}}_3$.

Keywords

quantum algebra, categorification, 2-categories, affine quantum algebra

Mathematical Subject Classification

Primary: 17B67

Secondary: 20G42

Milestones

Received: 17 October 2024

Revised: 18 September 2025

Accepted: 20 October 2025

Published: 10 March 2026