Multiplicative vertex algebras and quantum loop algebras

TL;DR

This paper introduces a multiplicative version of vertex coalgebras and demonstrates their compatibility with K-theoretic Hall algebras, linking them to quantum loop algebras.

Contribution

It defines multiplicative vertex coalgebras and shows their compatibility with equivariant K-theoretic Hall algebras, including the preprojective KHA.

Findings

Preprojective KHA admits a compatible multiplicative vertex coalgebra structure.

The structure is conjecturally isomorphic to certain quantum loop algebras.

Provides a new algebraic framework connecting Hall algebras and quantum groups.

Abstract

We define a multiplicative version of vertex coalgebras and show that various equivariant K-theoretic constructions of Hall algebras (KHAs) also admit a compatible multiplicative vertex coalgebra structure. In particular, this is true of the preprojective KHA of Varagnolo--Vasserot, which is (conjecturally) isomorphic to certain quantum loop algebras.