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 is defined as cohomology of the Artin moduli stack of representations with dimension vector. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can rene 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 ZI ≫0 × Z-graded vector space of the form V = V prim ⊗ℚ[x], where x is a variable of bidegree (0, 2) η Z I ≫0 × Z, and all the spaces ⊗kηZ V primyk, 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 (y, k) for which V primy,k 6= 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 suffciently 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]]. © 2012 Copyright Foundation Compositio Mathematica.