2-categorical affine symmetries of quantum enveloping algebras

TL;DR

This paper constructs 2-representations of affine quantum enveloping algebras' positive parts, extending known representations and relating to evaluation morphisms, with implications for character formulas.

Contribution

It introduces new 2-representations for affine quantum groups, extending existing frameworks and establishing connections to evaluation morphisms and prefundamental representations.

Findings

Constructed 2-representations for affine quantum groups in type A_n.

Extended the right-multiplication 2-representation to affine cases.

Provided new proofs for prefundamental representation character formulas.

Abstract

We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of $U_q^+(\mathfrak{sl}_{n+1})$ on itself and are closely related to evaluation morphisms of quantum groups. We expect that our 2-representation exists in all simple types and show that the corresponding 1-representation exists in types $D_4$ and $C_2$. We also show that a certain quotient of our 1-representation in type $A_n$ is isomorphic to a prefundamental representation. We use this to provide a new proof of the prefundamental representation character formulas in these cases.