Cohomological Hall algebra of a symmetric quiver

Efimov, Alexander I.

doi:10.1112/s0010437x12000152

Cited by 59 publications

(105 citation statements)

References 2 publications

Supporting

Mentioning

101

Contrasting

Order By: Relevance

“…, 0) is generic, and there is an equality A ζ θ=0 (Q) = H(M(Q), Q) vir . Under these conditions, Efimov [Efi12] proved that A ζ θ=0 (Q) ∼ = Sym (V prim ⊗ H(pt /C * , Q) vir ) for some V prim ∈ Vect + K such that each V prim,v has finite-dimensional total cohomology. Theorem 6.6 gives a precise definition of V prim in this case.…”

Section: 22mentioning

confidence: 99%

“…We hope that for the audience coming from the theory of cluster algebras, the absence of potentials and the resulting absence of arguments involving vanishing cycles and monodromic motives/mixed Hodge modules from e.g. [Nag13,Efi12,Dav18] will make this paper more approachable. Using Theorem 1.8, we can prove preservation of positivity for all four infinite families of 2-wall base cases (cf.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

View full text Add to dashboard Cite

Abstract. Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified "no exotics" type theorem for cohomological Donaldson-Thomas invariants, which we discuss and prove in the appendix.

show abstract

Section: 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

View full text Add to dashboard Cite

show abstract

“…A quiver Q is an oriented graph with a finite set of vertices Q 0 and a finite number of arrows between vertices α∶i → j. On ZQ 0 , we define the Euler form of Q by hd; ei Q ¼ P 17,[31][32][33].…”

Section: Moduli Of Quiver Representationsmentioning

confidence: 99%

BPS states, knots, and quivers

Kucharski

Reineke

Stošić³

et al. 2017

Phys. Rev. D

View full text Add to dashboard Cite

We argue how to identify the supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot, we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants (once the corresponding quiver is identified), conjectured originally based on string theory and M-theory arguments.

show abstract

“…For the symmetric quiver Q , it is conjectured by Kontsevich and Soibelman [8] and proved by Efimov [5] that the (Z I ≥0 , Z)-graded algebra (H, * ) is a free super-commutative algebra generated by a (Z I ≥0 , Z)-graded vector space V of the…”

Section: Introductionmentioning

confidence: 99%

“…Efimov [5], we can twist the multiplication by a sign such that (H, * ) is a super-commutative algebra with respect to the Z-grading. In fact, for a γ ,k ∈ H γ ,k , a γ ,k ∈ H γ ,k , we have:…”

Section: Introductionmentioning

confidence: 99%

Geometric construction of generators of CoHA of doubled quiver

Chen

2014

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

Presented by Claire VoisinLet Q be the double of a quiver. According to Efimov, Kontsevich and Soibelman, the cohomological Hall algebra (CoHA) associated with Q is a free super-commutative algebra. In this short note, we confirm a conjecture of Hausel, which gives a geometric realisation of the generators of the CoHA. © 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Soit Q le double d'un carquois. Selon Efimov, Kontsevich et Soibelman, l'algèbre cohomologique de Hall (CoHA) associée à Q est une algèbre libre super-commutative. Dans cette note, nous démontrons la conjecture de Hausel, donnant une réalisation géométrique des générateurs de cette algèbre.

show abstract

Cohomological Hall algebra of a symmetric quiver

Abstract: , 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

Cited by 59 publications

References 2 publications

Positivity for quantum cluster algebras

Positivity for quantum cluster algebras

BPS states, knots, and quivers

Geometric construction of generators of CoHA of doubled quiver

Contact Info

Product

Resources

About