Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics

1996 ◽  
Vol 82 (3) ◽  
pp. 473-502 ◽  
Author(s):  
Arkady Berenstein ◽  
Andrei Zelevinsky
Keyword(s):  
Quantum Group ◽  
Piecewise Linear ◽  
Canonical Bases

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 .

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.

Chaos in a manifold piecewise linear system

1981 ◽  
Vol 64 (10) ◽  
pp. 9-17 ◽  
Author(s):  
Toshimichi Saito ◽  
Hiroichi Fujita
Keyword(s):  
Linear System ◽  
Piecewise Linear ◽  
Piecewise Linear System
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