Cohomological Hall algebra of a symmetric quiver

Abstract

AbstractIn [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 quiverQwith a set of verticesIthe so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγis defined as cohomology of the Artin moduli stack of representations with dimension vectorγ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥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 ℤI≥0×ℤ-graded vector space of the formV=Vprim⊗ℚ[x] , wherexis a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁k∈ℤVprimγ,k,γ∈ℤI≥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 whichVprimγ,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 form-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].