Automorphisms of \(\check U_ q({\mathfrak b}^ +)\)
type: Item
instance of: scholarly article
title: Automorphisms of \(\check U_ q({\mathfrak b}^ +)\)
zbMATH Open document ID: 0888.17004
author: Odile Fleury
published in: Beiträge zur Algebra und Geometrie
publication date: 1997-10-06T00:00:00Z
full work available at URL: https://eudml.org/doc/225121
full work available at URL: no.2
review text: Let \(\mathfrak g\) be a simple Lie algebra over \(\mathbb C\) and \(q\) a nonzero complex number that is not a root of unity. The author computes the groups of algebra and Hopf algebra automorphisms of the augmented form \(\check U_q(\mathfrak b^+)\) of the quantized enveloping algebra of \(\mathfrak b^+\) with \(\mathfrak b^+\) a Borel subalgebra of \(\mathfrak g\). There are substantially fewer of these than there are automorphisms of \(U_q(\mathfrak g)\) or \(U(\mathfrak g)\). More precisely, it turns out the algebra automorphisms of \(\check U_q(\mathfrak b^+)\) are generated by a \(2n\)-dimensional multiplicative torus, the Dynkin diagram automorphisms, and the vector space of \(n\times n\) symmetric integer matrices, where \(n\) is the rank of \(\mathfrak g\). The Hopf algebra automorphisms are even simpler, being generated by an \(n\)-dimensional multiplicative torus and the Dynkin diagram automorphisms. By contrast, even the automorphism group of \(U(\mathfrak {sl}_2)\) is still unknown.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1066104
zbMATH Keywords: automorphisms
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: augmented form
zbMATH Keywords: Borel subalgebra
zbMATH Keywords: Hopf algebra
reviewed by: William M. McGovern
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1066104