Generators for K-theoretic Hall algebras of quivers with potential

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 Poincaré–Birkhoff–Witt theorem. We construct semi-orthogonal decompositions of categorical Hall algebras using techniques developed by Halpern-Leistner, Ballard–Favero–Katzarkov, and Špenko–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.