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

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+.

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.

Gröbner-Shirshov Basis of Quantum Groups

2015 ◽  
Vol 22 (03) ◽  
pp. 495-516 ◽  
Author(s):  
Gulshadam Yunus ◽  
Zhenzhen Gao ◽  
Abdukadir Obul
Keyword(s):  
Quantum Groups ◽  
Hall Algebra ◽  
Dynkin Type ◽  
Algebra Method

In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal Gröbner-Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.

scholarly journals A relation between a conjecture of Kac and the structure of the Hall algebra

2001 ◽  
Vol 160 (2-3) ◽  
pp. 319-332 ◽  
Author(s):  
Bert Sevenhant ◽  
Michel Van Den Bergh
Keyword(s):  
Hall Algebra

scholarly journals Drinfeld realization of the elliptic Hall algebra

2011 ◽  
Vol 35 (2) ◽  
pp. 237-262 ◽  
Author(s):  
Olivier Schiffmann
Keyword(s):  
Hall Algebra
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