On cohomological and K-theoretical Hall algebras of symmetric quivers

Abstract

We give a brief review of the cohomological Hall algebra CoHA 71 and the K-theoretical Hall algebra KHA 7Z associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) 7Z -+ 71 sigma similar to where 71 sigma similar to is a Zhang twist of the completion of 71. Moreover, we establish the equivalence of categories of "locally finite" graded modules 71- Modlf ⠄ 7ZQ- Modlf. Examples of locally finite 71-, resp. 7ZQ-modules