Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts

TL;DR

This paper constructs a vertex coproduct on the CoHA of quivers with potential, showing it forms a vertex bialgebra and connecting it to Drinfeld's deformed coproduct on Yangians, with implications for cohomological integrality.

Contribution

It introduces a new vertex coproduct on the Kontsevich--Soibelman CoHA, extending it to a vertex bialgebra and relating it to known structures like Yangians and previous coproducts.

Findings

Vertex coproduct forms a vertex bialgebra.

Recovers Drinfeld's deformed coproduct for ADE quivers.

Provides a new proof of the cohomological integrality theorem.

Abstract

We construct a vertex coproduct on the Kontsevich--Soibelman cohomological Hall algebra (CoHA) of a quiver with potential, following Joyce (2018). We show it forms a vertex bialgebra. By applying a vertex algebraic analogue of Majid--Radford bosonisation, we form an extension of the CoHA of quivers with potential which incorporates a Cartan part. In the case of ADE quivers our vertex coproduct recovers Drinfeld's deformed coproduct on the Yangian. We compare the vertex coproduct with a localised coproduct defined by Davison and with the construction of Dotsenko--Mozgovoy when the potential is trivial. Our construction gives a new proof of the cohomological integrality theorem for symmetric quivers with trivial potential.