Representations of quantum groups at even roots of unity
type: Item
instance of: scholarly article
title: Representations of quantum groups at even roots of unity
zbMATH Open document ID: 0812.17011
author: Jonathan Beck
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: Let \(U_ q({\mathfrak g})\) be the quantum enveloping algebra of the finite dimensional complex simple Lie algebra \({\mathfrak g}\) where \(q\) is a primitive \(\ell\)-th root of unity. When \(\ell\) is odd, the representation theory of \(\text{Spec } U_ q({\mathfrak g})\), the finite-dimensional irreducible representations was determined by \textit{C. De Concini} and \textit{V. Kac} [Colloq. Dixmier, Prog. Math. 92, 471-506 (1990; Zbl 0738.17008)], who partitioned \(\text{Spec } U_ q({\mathfrak g})\) by conjugacy classes of the corresponding algebraic group \(G\). The author of the paper under review handles the case when \(\ell\) is even, especially when \(\ell\) is divisible by 4. Here the partition goes by conjugacy classes in the group \(G^ \vee\), the Langlands dual of \(G\), and is more complicated than in the odd case. There is a discussion of a quantum coadjoint action and triangulability, and the degree of \(U_ q({\mathfrak g})\) is calculated by a deformation argument.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 614197
zbMATH Keywords: even roots of unity
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum enveloping algebra
zbMATH Keywords: quantum coadjoint action
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: jabr.1994.1174
OpenAlex ID: W2008140469
DOI: 10.1006/JABR.1994.1174
description: scientific article; zbMATH DE number 614197