Cohomological Hall algebra of a symmetric quiver

Abstract

In [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤ^I_≥0-graded. Its graded component ℋ_γ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤ^I_≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤ^I_≥0×ℤ-graded vector space of the form V =V^prim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤ^I_≥0×ℤ, and all the spaces ⨁ _k∈ℤV^prim_γ,k, γ∈ℤ^I_≥0. 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≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].