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 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.
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Acknowledgements
I would like to thank my PhD advisor Davesh Maulik for suggesting the problem discussed in the present paper and for his constant help and encouragement throughout the project. I would like to thank Ben Davison, Pavel Etingof, Daniel Halpern-Leistner, Andrei Neguţ, Andrei Okounkov, Yukinobu Toda, and Eric Vasserot for useful conversations about the project. I thank the referee for many useful comments.
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Pădurariu, T. Generators for K-theoretic Hall algebras of quivers with potential. Sel. Math. New Ser. 30, 4 (2024). https://doi.org/10.1007/s00029-023-00891-6
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DOI: https://doi.org/10.1007/s00029-023-00891-6