The Critical CoHA of a Quiver With Potential

Abstract

Pursuing the similarity between the Kontsevich--Soibelman construction of the cohomological Hall algebra of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the inetgrality conjecture for motivic Donaldson--Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the cohomological Hall algebra associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of cohomological Hall algebras with Yangians, and explaining how to connect the study of character varieties with the study of cohomological Hall algebras.