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We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the “dimensional reduction” from 3-dimensional Calabi–Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson–Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi–Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi–Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).

progress in mathematics

Cohomological Hall Algebras, Semicanonical Bases and Donaldson–Thomas Invariants for 2-dimensional Calabi–Yau Categories (with an Appendix by Ben Davison)