Quantum determinantal ideals
type: Item
instance of: scholarly article
title: Quantum determinantal ideals
zbMATH Open document ID: 0958.16025
DOI: 10.1215/S0012-7094-00-10318-3
published in: Duke Mathematical Journal
publication date: 2001-04-03T00:00:00Z
review text: The quantized coordinate ring \({\mathcal O}_q(M_n(k))\) of \(n\times n\) matrices over the field \(k\) is an affine \(k\)-algebra whose relations reduce, when the parameter \(q\) is \(1\), to ones which define the commutative polynomial \(k\)-algebra in \(n^2\) indeterminates -- that is, the classical coordinate ring of \(n\times n\) matrices over \(k\). \({\mathcal O}_q(M_n(k))\) is a bialgebra with the same coproduct as in the classical case, and contains a non-zero central element \(D\), the quantum determinant, such that \({\mathcal O}_q(M_n(k))/\langle D-1\rangle={\mathcal O}_q(\text{SL}_n(k))\) and \({\mathcal O}_q(M_n(k))[D^{-1}]={\mathcal O}_q(\text{GL}_n(k))\) are Hopf algebras, respectively quantum \(\text{SL}_n(k)\) and quantum \(\text{GL}_n(k)\). The prime and primitive spectra of these latter two Hopf algebras have been intensively studied -- see [\textit{A. Joseph}, Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3) 29, Berlin, Springer (1995; Zbl 0808.17004)] and the references there. The purpose of this paper is to begin the study of that part of the prime spectrum of \({\mathcal O}_q(M_n(k))\) consisting of ideals containing \(D\), about which the earlier work can give no information. The main result of this paper is a quantum analogue of the first fundamental theorem of invariant theory. It states that, for \(t\) an integer between \(0\) and \(n-1\), the ideal of \({\mathcal O}_q(M_n(k))\) generated by the \((t+1)\times(t+1)\) quantum minors of \({\mathcal O}_q(M_n(k))\) is completely prime. The ideas developed for its proof are likely to find further application in the analysis of \({\mathcal O}_q(M_n(k))\), the main technical tool being the construction of a suitable vector space basis of \({\mathcal O}_q(M_n(k))\), an analogue of the standard basis in the commutative case, such that \({\mathcal O}_q(M_n(k))\) can be viewed as a ``noncommutative algebra with straightening law'' with respect to this basis.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W35
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 1463083
zbMATH Keywords: noncommutative algebras with straightening laws
zbMATH Keywords: quantized coordinate rings
zbMATH Keywords: quantum determinants
zbMATH Keywords: Hopf algebras
zbMATH Keywords: primitive spectra
zbMATH Keywords: prime spectra
zbMATH Keywords: invariant theory
zbMATH Keywords: quantum minors
zbMATH Keywords: vector space bases
author: Kenneth R. Goodearl
author: T. H. Lenagan
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Quantum deformations of GLn
cites work: Prime spectra of quantum semisimple groups
cites work: Determinantal rings
cites work: Young diagrams and determinantal varieties
cites work: The first fundamental theorem of coinvariant theory for the quantum general linear group
cites work: Primitive ideals of \(\mathbb{C}_ q[SL(n)]\)
cites work: Standard basis theorem for quantum linear groups
cites work: Q4317871
cites work: The quantum coordinate ring of the special linear group
cites work: Q3998175
cites work: Q4280722
cites work: Quantum linear groups
cites work: Normalité de certains anneaux déterminantiels quantiques
full work available at URL: s0012-7094-00-10318-3
OpenAlex ID: W1968543247
description: scientific article; zbMATH DE number 1463083