# A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver

## TL;DR

This paper introduces a family of explicit isomorphisms for the cohomological Hall algebra of acyclic quivers, revealing new decompositions linked to Dynkin subquivers and providing topological realizations of known algebraic structures.

## Contribution

It constructs a family of structure isomorphisms for CoHAs of acyclic quivers, generalizing and explicitly realizing previous decomposition results.

## Key findings

- Decomposition of CoHA into tensor products related to Dynkin subquivers.
- Explicit topological realization of algebraic isomorphisms.
- Connection between structure constants and Dynkin quiver polynomials.

## Abstract

For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The family is parameterized by partitions of the quiver into Dynkin subquivers. For each such partition, we write the domain of our isomorphism as a tensor product of subalgebras in two ways. In the first, each tensor factor is isomorphic to the CoHA of the quiver with a single vertex and no arrows. In the second, the tensor factors are each isomorphic to the CoHAs of the corresponding Dynkin subquivers. When the quiver is already an orientation of a simply-laced Dynkin diagram, our results interpolate between isomorphisms proved by Rimanyi. Such CoHA decompositions appear in prior work of Davison--Meinhardt and Franzen--Reineke, but our proof gives an explicit topological realization of these results. As a consequence of our method, we deduce that certain structure constants in the CoHA naturally arise as CoHA products of classes of Dynkin quiver polynomials.