Generators for K-theoretic Hall algebras of quivers with potential

TL;DR

This paper constructs semi-orthogonal decompositions of categorical K-theoretic Hall algebras for quivers with potential, proving a PBW-type theorem and identifying generators via intersection K-theory of moduli spaces.

Contribution

It develops new semi-orthogonal decompositions for categorical Hall algebras and establishes a PBW theorem for their quotients, advancing understanding of K-theoretic Hall algebra structures.

Findings

Constructed semi-orthogonal decompositions using advanced techniques.

Proved a PBW-type theorem for quotients of K-theoretic Hall algebras.

Identified generators via intersection K-theory of moduli spaces.

Abstract

K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are a generalization of preprojective KHAs of quivers, which are conjecturally positive parts of the Okounkov-Smironov quantum affine algebras. In particular, preprojective KHAs are expected to satisfy a Poincar\'e-Birkhoff-Witt theorem. We construct semi-orthogonal decompositions of categorical Hall algebras using techniques developed by Halpern-Leistner, Ballard-Favero-Katzarkov, and \v{S}penko-Van den Bergh. For a quotient of $\text{KHA}(Q,W)_{\mathbb{Q}}$, we refine these decompositions and prove a PBW-type theorem for it. The spaces of generators of $\text{KHA}(Q,0)_{\mathbb{Q}}$ are given by (a version of) intersection K-theory of coarse moduli spaces of representations of $Q$.