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K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are ageneralization of preprojective KHAs of quivers, which are conjecturallypositive 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 usingtechniques developed by Halpern-Leistner, Ballard-Favero-Katzarkov, and\v{S}penko-Van den Bergh. For a quotient of $\text{KHA}(Q,W)_{\mathbb{Q}}$, werefine these decompositions and prove a PBW-type theorem for it. The spaces ofgenerators of $\text{KHA}(Q,0)_{\mathbb{Q}}$ are given by (a version of)intersection K-theory of coarse moduli spaces of representations of $Q$.

Generators for K-theoretic Hall algebras of quivers with potential