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 $\cH,$ which is $\Z_{\geq 0}^I$-graded. Its graded component $\cH_γ$ is defined as cohomology of Artin moduli stack of representations with dimension vector $γ.$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to $\Z_{\geq 0}^I\times\Z,$ and modify the product by a sign to get a super-commutative algebra $(\cH,\star)$ (with parity induced by $\Z$-grading). It is conjectured in \cite{KS} that in this case the algebra $(\cH\otimes\Q,\star)$ is free super-commutative generated by a $\Z_{\geq 0}^I\times\Z$-graded vector space of the form $V=V^{prim}\otimes\Q[x],$ where $x$ is a variable of bidegree $(0,2)\in\Z_{\geq 0}^I\times\Z,$ and all the spaces $\bigoplus\limits_{k\in\Z}V^{prim}_{γ,k},$ $γ\in\Z_{\geq 0}^I.$ are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs $(γ,k)$ for which $V^{prim}_{γ,k}\ne 0$ (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}.