scholarly journals On cohomological Hall algebras of quivers: Generators

2020 ◽  
Vol 2020 (760) ◽  
pp. 59-132 ◽  
Author(s):  
Olivier Schiffmann ◽  
Eric Vasserot
Keyword(s):  
Constant Term ◽  
Hall Algebra ◽  
Preprojective Algebra ◽  
Quiver Varieties ◽  
Hall Algebras ◽  
Root Multiplicities ◽  
Moduli Stack

AbstractWe study the cohomological Hall algebra {\operatorname{Y}\nolimits^{\flat}} of a Lagrangian substack {\Lambda^{\flat}} of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that {\operatorname{Y}\nolimits^{\flat}} is pure and we compute its Poincaré polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that {\operatorname{Y}\nolimits^{\flat}} is equal, after a suitable extension of scalars, to the Yangian {\mathbb{Y}} introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov’s conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac–Moody algebras.

scholarly journals On two cohomological Hall algebras

2019 ◽  
Vol 150 (3) ◽  
pp. 1581-1607
Author(s):  
Yaping Yang ◽  
Gufang Zhao
Keyword(s):  
Equivariant Cohomology ◽  
The Other ◽  
Algebra Homomorphism ◽  
Hall Algebra ◽  
Preprojective Algebra ◽  
Positive Part ◽  
Hall Algebras ◽  
Oriented Cohomology ◽  
Quiver With Potential ◽  
Vanishing Cycles

AbstractWe compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced in [19] associated with each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot [15]. The other one is the critical CoHA defined by Kontsevich-Soibelman associated with each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients in the sheaf of vanishing cycles. In the present paper, we show that the critical CoHA, for the quiver with potential of Ginzburg, is isomorphic to the preprojective CoHA as algebras. As applications, we obtain an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

scholarly journals Spherical Hall Algebras of Weighted Projective Curves

2018 ◽  
Vol 2020 (15) ◽  
pp. 4721-4775
Author(s):  
Jyun-Ao Lin
Keyword(s):  
Vector Bundles ◽  
Characteristic Functions ◽  
Projective Curve ◽  
Hall Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Hall Algebras ◽  
Arbitrary Genus ◽  
Projective Curves ◽  
Fixed Rank

Abstract In this article, we deal with the structure of the spherical Hall algebra $\mathbf{U}$ of coherent sheaves with parabolic structures on a smooth projective curve $X$ of arbitrary genus $g$. We provide a shuffle-like presentation of the bundle part $\mathbf{U}^>$ and show the existence of generic spherical Hall algebra of genus $g$. We also prove that the algebra $\mathbf{U}$ contains the characteristic functions on all the Harder–Narasimhan strata. These results together imply Schiffmann’s theorem on the existence of Kac polynomials for parabolic vector bundles of fixed rank and multi-degree over $X$. On the other hand, the shuffle structure we obtain is new and we make links to the representations of quantum affine algebras of type $A$.

scholarly journals Cohomological Hall algebras and character varieties

2016 ◽  
Vol 27 (07) ◽  
pp. 1640003
Author(s):  
Ben Davison
Keyword(s):  
Hall Algebra ◽  
Hall Algebras ◽  
Character Varieties ◽  
Specific Instance ◽  
The Relationship

In this paper, we investigate the relationship between twisted and untwisted character varieties, via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi–Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson–Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the [Formula: see text] polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.

scholarly journals Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

2019 ◽  
Author(s):  
Haicheng Zhang
Keyword(s):  
Finite Field ◽  
Path Algebra ◽  
Hall Algebra ◽  
Minimal Set ◽  
Hall Algebras ◽  
Enveloping Algebra ◽  
Pbw Basis ◽  
Perfect Complexes ◽  
Dynkin Quiver ◽  
Minimal Generators

Abstract Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra ${\mathcal{H}}\,(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations among the generators in ${\mathcal{H}}\,(C_1(\mathscr{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of ${\mathcal{H}}\,(C_1(\mathscr{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathscr{P})$, respectively.

scholarly journals Legendrian skein algebras and Hall algebras

2021 ◽  
Author(s):  
Fabian Haiden
Keyword(s):  
Product Type ◽  
Natural Homomorphism ◽  
Fukaya Category ◽  
Hall Algebra ◽  
Associative Algebras ◽  
Quantum Topology ◽  
Hall Algebras ◽  
Legendrian Links ◽  
Skein Algebras ◽  
Marked Points

AbstractWe compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type $$S\times {\mathbb {R}}$$ S × R . The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus.

scholarly journals Quiver varieties and Hall algebras

2016 ◽  
Vol 112 (6) ◽  
pp. 1002-1018 ◽  
Author(s):  
Sarah Scherotzke ◽  
Nicolo Sibilla
Keyword(s):  
Quiver Varieties ◽  
Hall Algebras

scholarly journals On Constructing Ideals of the Hall Algebra of Type B

2010 ◽  
Vol 17 (01) ◽  
pp. 47-58
Author(s):  
Qunhua Liu
Keyword(s):  
Algebra Homomorphism ◽  
Type B ◽  
Hall Algebra ◽  
Schur Algebra ◽  
Hall Algebras ◽  
Dynkin Quivers

Let Hv(An) and Hv(Bn) be the Hall algebras over ℚ(v) of the Dynkin quivers An and Bn (n ≥ 1), respectively, where v is an indeterminate and the quivers have linear orientation. By comparing the quantum Serre relations, we find a natural algebra epimorphism π : Hv(Bn) → Hv2(An). We determine the kernel of π by giving two sets of generators. Let φ be the natural algebra homomorphism from Hv(An) to the quantized Schur algebra Sv(n + 1, r)(r ≥ 1) and write [Formula: see text] for the induced map. We obtain several ideals of Hv(Bn) by lifting the kernel of φ to the kernel of the composition map [Formula: see text].

scholarly journals THE HALL ALGEBRAS OF SURFACES I

2018 ◽  
Vol 19 (3) ◽  
pp. 971-1028 ◽  
Author(s):  
Benjamin Cooper ◽  
Peter Samuelson
Keyword(s):  
Explicit Description ◽  
Fukaya Category ◽  
Hall Algebra ◽  
Hall Algebras ◽  
Skein Relation

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with $m$ marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.

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