The center of the quantized matrix algebra
type: Item
instance of: scholarly article
title: The center of the quantized matrix algebra
zbMATH Open document ID: 0886.17015
author: H. P. Jakobsen
author: Zhang Hechun
published in: Journal of Algebra
publication date: 1998-05-04T00:00:00Z
full work available at URL: the-center-of-the-quantized-matrix-algebra(ca47e530-6c38-11dc-bee9-02004c4f4f50).html
review text: \textit{M. Noumi, H. Yamada} and \textit{K. Mimachi} [Jap. J. Math., New Ser. 19, No. 1, 31-80 (1993; Zbl 0806.17016)] have proved that the center of the quantized matrix algebra \(M_q(n)\) is generated by the quantum determinant if \(q\) is generic (i.e., \(q\) is not a root of unity). In the paper under review the authors provide a set of generators for the center of \(M_q(n)\) at any primitive root of unity. In order to show that the \(m\)-th powers of the generators of \(M_q(n)\) (where \(m\) is the degree of \(q\)) and certain elements derived from the quantum determinant indeed generate the center, they determine the degree of \(M_q(n)\). By results of \textit{C. De Concini} and \textit{C. Procesi} [Lect. Notes Math. 1565, 31-140 (1993; Zbl 0795.17005)], the degree of \(M_q(n)\) coincides with the degree of its associated quasi-polynomial algebra \(\overline {M_q(n)}\), and the latter as well as a basis for the center of \(\overline {M_q(n)}\) are given in terms of a skew-symmetric integer matrix considered as a linear map modulo \(m\). From this together with properties of certain restricted quantized Verma modules the authors obtain the degree and a set of generators for the center of \(\overline {M_q(n)}\) and \(M_q(n)\). As it is common for quantum groups, the details are easier if the degree of \(q\) is odd (and for the center only in this case an explicit proof is given in the paper). The results of this paper are related to similar results for quantized function algebras of simply connected semisimple complex algebraic groups at roots of unity, where \textit{G. Gaiffi} [J. Algebra 180, No. 3, 897-917 (1996; Zbl 0862.17009)] determined the degree and \textit{B. Enriquez} [Bull. Soc. Math. Fr. 122, No. 4, 433-485 (1994; Zbl 0851.17018)] [see also the appendix to Adv. Math. 108, No. 2, 205-262 (1994; Zbl 0846.17008)] obtained a description of the center.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1088592
zbMATH Keywords: quantized matrix algebra
zbMATH Keywords: quantum determinant
zbMATH Keywords: root of unity
zbMATH Keywords: center
zbMATH Keywords: degree
zbMATH Keywords: restricted quantized Verma modules
MaRDI profile type: Publication
OpenAlex ID: W2013534658
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DOI: 10.1006/JABR.1997.7121
description: scientific article; zbMATH DE number 1088592