Cohomological Hall algebra of a symmetric quiver

In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological Hall algebra $\text{cH},$ which is ${\mathbb{Z}}_{\ge 0}^I$-graded. Its graded component ${\text{cH}}_{\gamma }$ is defined as cohomology of Artin moduli stack of representations with dimension vector $\gamma \mathrm{.}$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to ${\mathbb{Z}}_{\ge 0}^I\times \mathbb{Z},$ and modify the product by a sign to get a super-commutative algebra $(\text{cH},\star )$ (with parity induced by $\mathbb{Z}$-grading). It is conjectured in \cite{KS} that in this case the algebra $(\text{cH}\otimes \text{Q},\star )$ is free super-commutative generated by a ${\mathbb{Z}}_{\ge 0}^I\times \mathbb{Z}$-graded vector space of the form $V=V^{prim}\otimes \text{Q}[x],$ where $x$ is a variable of bidegree $(0,2)\in\Z_{\geq 0}^I\times\Z,$andallthespaces$\bigoplus\limits_{k\in\Z}V^{prim}_{\gamma,k},$ $\gamma \in {\mathbb{Z}}_{\ge 0}^I\mathrm{.}$ are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs $(\gamma ,k)$ for which $V_{\gamma ,k}^{prim}\mathrm{\ne }0$ (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson-Thomas invariants, which was used by S. Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops \cite{M}. Finally, we mention a connection with the paper of Reineke \cite{R}.