its cohomological Hall algebra (CoHA) ${\mathscr{H}}(Q) = {\mathscr{H}}$, which Kontsevich and Soibelman defined in their seminal paper [@mkys2011]. The CoHA is inspired by physics; it is designed to model the algebra of BPS states in string theory. Mathematically speaking, let $\gamma$ denote a dimension vector for $Q$. As a vector space, the CoHA is a direct sum over all dimension vectors ${\mathscr{H}}= {\bigoplus}_\gamma {\mathscr{H}}_\gamma$. The summands are the equivariant cohomology algebras ${\mathscr{H}}_\gamma = {\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_\gamma}({\boldsymbol{\mathrm{Rep}}}_\gamma)$ where ${\boldsymbol{\mathrm{Rep}}}_\gamma$ is the space of $Q$-representations of dimension $\gamma$ and ${\boldsymbol{{\mathrm{GL}}}}_\gamma$ is the algebraic group which acts naturally on ${\boldsymbol{\mathrm{Rep}}}_\gamma$ by simultaneously changing basis at each vertex. The novel multiplication (generally noncommutative) endowed by Kontsevich–Soibelman on the CoHA, and which we denote by $*$ throughout the paper, respects the dimension vector grading and encodes important combinatorial, algebraic, and geometric aspects of the quiver and its representations, see [@mkys2011; @ae2012; @rr2013; @bd2017; @hfmr2018].

The structure of ${\mathscr{H}}$ (and its modules) has been the subject of much study [@mkys2011; @ae2012; @rr2013; @zc2014; @bd2017; @hf2016; @hf2018; @hfmr2018; @hfmr2019]. In this paper, we restrict to the case of quivers $Q$ which are acyclic, i.e., have no oriented cycles. Given a quiver, one problem is to determine subalgebras over which 