scholarly journals The cohomological Hall algebra of a preprojective algebra

2018 ◽  
Vol 116 (5) ◽  
pp. 1029-1074 ◽  
Author(s):  
Yaping Yang ◽  
Gufang Zhao
Keyword(s):  
Hall Algebra ◽  
Preprojective Algebra

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 Colored Trees and Noncommutative Symmetric Functions

10.37236/468 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Matt Szczesny
Keyword(s):  
Hopf Algebra ◽  
Symmetric Functions ◽  
Hall Algebra ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions

Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

Generic Extensions and Canonical Bases for Cyclic Quivers

2007 ◽  
Vol 59 (6) ◽  
pp. 1260-1283 ◽  
Author(s):  
Bangming Deng ◽  
Jie Du ◽  
Jie Xiao
Keyword(s):  
Hall Algebra ◽  
Quiver Representations ◽  
Algebraic Construction ◽  
Positive Part ◽  
Monomial Basis ◽  
Canonical Bases ◽  
Generic Extensions ◽  
Cyclic Quiver

AbstractWe use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the positive part U+of the quantum affine. This construction relies on analysis of quiver representations and the introduction of a new integral PBW–like basis for the Lusztig ℤ[v,v–1]-form of U+.

Gröbner–Shirshov Basis of Derived Hall Algebra of Type An

2020 ◽  
Vol 36 (8) ◽  
pp. 929-942
Author(s):  
Zhe He ◽  
Abdukadir Obul
Keyword(s):  
Hall Algebra
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