Categorical and K-theoretic Hall algebras for quivers with potential

TL;DR

This paper develops a categorification of Hall algebras for quivers with potential, connecting them to Yangians and quantum affine algebras through K-theoretic and geometric methods.

Contribution

It constructs a categorified Hall algebra using categories of singularities and establishes a wall-crossing theorem, linking to quantum groups and Yangians.

Findings

Constructed a categorification of Hall algebras for quivers with potential.

Proved a wall-crossing theorem for K-theoretic Hall algebras.

Expected to recover the positive part of quantum affine algebras.

Abstract

Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the critical cohomology of the stack of representations of $(Q,W)$. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann-Vasserot, Maulik-Okounkov, Yang-Zhao etc. about geometric constructions of Yangians and their representations; indeed, given a quiver $Q$, there exists an associated pair $\left(\widetilde{Q},\widetilde{W}\right)$ whose CoHA is conjecturally the positive half of the Maulik-Okounkov Yangian $Y_{\text{MO}}(\mathfrak{g}_Q)$. For a quiver with potential $(Q,W)$, we follow a suggestion of Kontsevich-Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers and we prove a wall-crossing theorem for KHAs. We expect the KHA for $\left(\widetilde{Q},\widetilde{W}\right)$ to recover the positive part of quantum affine algebra $U_q\left(\widehat{\mathfrak{g}_Q}\right)$ defined by Okounkov-Smirnov.