\,
        \stackrel{{\cong}}{\longrightarrow}
        {\mathscr{H}}.$$ In the case of orientations of type $A$, $D$, or $E$ Dynkin diagrams, the isomorphisms interpolate between Rimányi’s isomorphisms , but in general not all semistable CoHAs ${\mathscr{H}}^{\theta\text{-sst},\mu}$ are isomorphic to ${\mathscr{H}}(A_1)$. The algebras ${\mathscr{H}}^{\theta\text{-sst},\mu}$ do admit understood presentations even if explicit descriptions are difficult in particular cases [@hfmr2018 Section 8]. For example, Franzen and Reineke considered the application of to the so-called Kroneker quiver $1 \leftleftarrows 2$ [@hfmr2019].

Our main theorem, from which Corollaries \[cor:dynkin.decomp\] and \[cor:dynkin.orbit.mult\] follow, is the following decomposition result which, in the case when $Q$ is Dynkin, interpolates between the lefthand and righthand decompositions of .

[ **Theorem \[thm:main\]**.  *Let $\{Q^1,\ldots,Q^\ell\}$ be an admissible Dynkin subquiver partition of the acyclic quiver $Q$. If $r_j$ denotes the number of positive roots for $Q^j$ and $r=\sum_j r_j$, there exist subalgebras ${\mathscr{A}}_{1},\ldots,{\mathscr{A}}_{r} \subset {\mathscr{H}}(Q)$ such that each ${\mathscr{A}}_{u}$ is isomorphic to ${\mathscr{H}}(A_1)$, and such that the left to right $*$-multiplication induces an isomorphism $$\pushQED{\qed} {\mathscr{A}}_{1} {\otimes}\cdots {\otimes}{\mathscr{A}}_{r} {\longrightarrow\,} {\mathscr{H}}(Q). \qedhere \popQED$$* ]{}

In the context of the results of Franzen–Reineke and Davison–Meinhardt, we comment that the methods of our paper adopt a fundamentally *topological viewpoint*, whereas the stability condition approach is a fundamentally *representation theoretical 