left to right. Here, the passage to Poincaré series reproduces the quantum pentagon identity, going back to the work of Faddeev–Kashaev [@lfrk1994].

II\. Rimányi [@rr2013] proved that the two decompositions from generalized to the case when $Q$ is an orientation of any simply-laced Dynkin diagram (i.e., of type $A$, $D$, or $E$). In particular, there are subalgebras ${\mathscr{A}}_\beta$ for each positive root $\beta$ (and hence for every simple root) such that $$\label{eqn:intro.RR}
    \left({\bigotimes}^{\to}_{\text{$\alpha$ a simple root}} {\mathscr{A}}_\alpha \right)
    \stackrel{{\cong}}{\longrightarrow} {\mathscr{H}}\stackrel{{\cong}}{\longleftarrow} 
    \left({\bigotimes}^{\to}_{\text{$\beta$ a positive root}} {\mathscr{A}}_\beta \right)$$ where the arrows over the tensor products indicate the $*$-multiplications must be carried out in a certain prescribed order. Each tensor factor subalgebra in is isomorphic to ${\mathscr{H}}(A_1)$.

III\. More recently, works of Davison–Meinhardt [@bdsm2016] and Franzen–Reineke [@hfmr2018] give more refined tensor product decompositions which hold for any quiver. In the case of Davison–Meinhardt, the result generalizes completely to any so-called *quiver with potential* (for acyclic quivers, any potential must necessarily be zero). Their CoHA isomorphisms are parameterized by a fixed stability condition $\theta$ and the tensor product is over the associated semi-stable CoHAs (with potential) taken in descending order of slope $\mu$ (see [@hfmr2018 Theorem 6.1] and [@bdsm2016 Theorem D]) $$\label{eqn:intro.FR}
        {\bigotimes}^{\to}_{\mu} {\mathscr{H}}^{\theta\text{-sst},\mu} 