Cohomological Hall algebra of a symmetric quiver

Abstract

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 ${\mathbb{Z}}_{\ge 0}^I$-graded. Its graded component $\cH_γ$ is defined as cohomology of Artin moduli stack of representations with dimension vector $\gamma .$ 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 $(\cH,\star)$ (with parity induced by $\mathbb{Z}$-grading). It is conjectured in \cite{KS} that in this case the algebra $(\cH\otimes\Q,\star)$ is free super-commutative generated by a ${\mathbb{Z}}_{\ge 0}^I\times \mathbb{Z}$-graded vector space of the form $V=V^{prim}\otimes\Q[x],$ where $x$ is a variable of bidegree $(0,2)\in {\mathbb{Z}}_{\ge 0}^I\times \mathbb{Z},$ and all the spaces $\underset{k\in \mathbb{Z}}{⨁}{V}_{\gamma ,k}^{prim},$ $\gamma \in {\mathbb{Z}}_{\ge 0}^I.$ 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 $V_{\gamma ,k}^{prim}\ne 0$ (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}.

16 pages, no figures; a reference added

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