location:  Publications → journals → CJM
Abstract view

Multiplication Formulas and Canonical Bases for Quantum Affine $gl_n$

Published:2017-08-16
Printed: Aug 2018
• Jie Du,
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
• Zhonghua Zhao,
School of Science, Beijing University of Chemical Technology, Beijing 100029, China
 Format: LaTeX MathJax PDF

Abstract

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. 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 certain monomial base for $\mathfrak{H}_\Delta(n)$ given by Deng, Du, and Xiao together with the double Ringel--Hall algebra realisation of the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. 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 $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.
 Keywords: Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis
 MSC Classifications: 16G20 - Representations of quivers and partially ordered sets 20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

 top of page | contact us | privacy | site map |