our results in context. The structure of ${\mathscr{H}}$ for the quiver with one vertex and no loops, i.e., the Dynkin diagram of type $A_1$, was determined already in the original work of Kontsevich and Soibelman [@mkys2011 Section 2.5]. ${\mathscr{H}}(A_1)$ is an exterior algebra on countably many generators; we recount this example in Section \[ss:coha.A1\]. Efimov proved an analogous result, conjectured by Kontsevich–Soibelman, for any symmetric quiver [@ae2012]. Efimov found primitive generators for the CoHA of a symmetric quiver, and the description in this case established the positive integrality of the associated Donaldson–Thomas invariants. We highlight three other relevant points in this story (at times considering non-acyclic quivers).

I. Kontsevich and Soibelman also considered the quiver $1 \leftarrow 2$ (an orientation of the $A_2$ Dynkin diagram). They gave a decomposition of ${\mathscr{H}}(A_2)$ in two ways: first as a tensor product of subalgebras parameterized by simple roots of the corresponding root system, and second as a tensor product of subalgebras parameterized by the positive roots [@mkys2011 Section 2.8]. That is, they described subalgebras ${\mathscr{A}}_{1,0}$, ${\mathscr{A}}_{0,1}$, and ${\mathscr{A}}_{1,1}$ such that each is isomorphic to ${\mathscr{H}}(A_1)$ and furthermore $$\label{eqn:intro.A2} 
    {\mathscr{A}}_{1,0} {\otimes}{\mathscr{A}}_{0,1} 
    \stackrel{{\cong}}{\longrightarrow} {\mathscr{H}}(A_2) 
    \stackrel{{\cong}}{\longleftarrow} {\mathscr{A}}_{0,1} {\otimes}{\mathscr{A}}_{1,1} {\otimes}{\mathscr{A}}_{1,0}$$ when the $*$-multiplication is carried out from 