Research Article

Cohomological Hall algebra of a symmetric quiver

Alexander I. Efimov^a1a2

^a1 Steklov Mathematical Institute of RAS, Gubkin str. 8, GSP-1, Moscow 119991, Russia

^a2 Independent University of Moscow, Moscow, Russia (email: efimov@mccme.ru)

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∈ℤ V ^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]].

(Received May 13 2011)

(Accepted October 10 2011)

(Online publication May 15 2012)

2010 Mathematics Subject Classification

• 14F25;
• 14N35;
• 16G20 (primary)

Keywords

• Donaldson–Thomas invariants;
• cohomological Hall algebra;
• Kac’s conjecture

Footnotes

The author was partially supported by ‘Dynasty’ Foundation, Simons Foundation, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.