Braid group action and canonical bases
type: Item
instance of: scholarly article
title: Braid group action and canonical bases
zbMATH Open document ID: 0861.17008
author: G. Lusztig
published in: Advances in Mathematics
publication date: 1997-05-14T00:00:00Z
review text: The author shows that the action of the braid group on a quantized enveloping algebra \(U\) (possibly of infinite type) is well compatible with the canonical basis. More precisely, if \(U^+\) is the plus-part of \(U\) and we decompose \[ U^+ =\bigl(U^+ \cap T_i^{-1} (U^+) \bigr) \oplus U^+ E_i, \quad U^+ =\bigl(T_i(U^+) \cap U^+ \bigr) \oplus E_iU^+ \] then the projections of the canonical basis vectors of \(U^+\) onto the first factor which are not zero form a basis of \(U^+ \cap T_i^{-1} (U^+)\) and \(T_i(U^+) \cap U^+\) respectively, and under \(T_i\) these bases correspond. This result leads to a parametrization of a big part of the canonical basis.
reviewed by: Wolfgang Soergel
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 966464
zbMATH Keywords: quantum groups
zbMATH Keywords: braid group
zbMATH Keywords: canonical basis
MaRDI profile type: Publication
full work available at URL: aima.1996.0061
OpenAlex ID: W2068804561
DOI: 10.1006/AIMA.1996.0061
description: scientific article