Multiplication Formulas and Canonical Bases for Quantum Affine gln

2018 ◽  
Vol 70 (4) ◽  
pp. 773-803 ◽  
Author(s):  
Jie Du ◽  
Zhonghua Zhao
Keyword(s):  
Quantum Group ◽  
Canonical Basis ◽  
Recursive Formula ◽  
Loop Algebra ◽  
Hall Algebra ◽  
Monomial Basis ◽  
Multiplication Formula ◽  
Canonical Bases ◽  
Theoretic Proof ◽  
Cyclic Quiver

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

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 The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials. L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .

scholarly journals The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal

2010 ◽  
Vol Vol. 12 no. 5 (Combinatorics) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Type D ◽  
Dual Canonical Basis ◽  
International Audience

Combinatorics International audience The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.

scholarly journals The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

2018 ◽  
Vol 2019 (20) ◽  
pp. 6179-6215 ◽  
Author(s):  
Jie Du ◽  
Qiang Fu
Keyword(s):  
Canonical Basis ◽  
Alternative Algebra ◽  
Integral Form ◽  
Loop Algebra ◽  
Algebra Structure ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Weyl Duality ◽  
Integral Quantum

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

scholarly journals Canonical bases arising from quantum symmetric pairs of Kac–Moody type

2021 ◽  
Vol 157 (7) ◽  
pp. 1507-1537
Author(s):  
Huanchen Bao ◽  
Weiqiang Wang
Keyword(s):  
Quantum Group ◽  
Finite Type ◽  
Canonical Basis ◽  
Integral Form ◽  
New Approach ◽  
Canonical Bases ◽  
Highest Weight ◽  
Technical Assumption ◽  
Rank One ◽  
Divided Powers

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

scholarly journals Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

2019 ◽  
Vol 19 (03) ◽  
pp. 2050054
Author(s):  
Zhonghua Zhao
Keyword(s):  
Canonical Basis ◽  
Recursive Formula ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Hall Algebras ◽  
Pbw Basis ◽  
Loewy Length ◽  
Generic Extensions

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

scholarly journals Canonical bases and standard monomials

1998 ◽  
Vol 41 (3) ◽  
pp. 611-623
Author(s):  
R. J. Marsh
Keyword(s):  
Lie Algebra ◽  
Canonical Basis ◽  
Monomial Basis ◽  
Fundamental Weight ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Standard Monomials ◽  
Standard Monomial ◽  
Quantized Enveloping Algebra

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

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