Catenarity in quantum algebras
type: Item
instance of: scholarly article
title: Catenarity in quantum algebras
zbMATH Open document ID: 0864.16018
DOI: 10.1016/0022-4049(95)00120-4
author: Kenneth R. Goodearl
author: T. H. Lenagan
published in: Journal of Pure and Applied Algebra
publication date: 1997-06-12T00:00:00Z
review text: In a celebrated but never-published work [Equidimensionalité de la variété caractéristique, Exposé de O. Gabber rédigé par T. Levasseur, Université de Paris VI (1982)] \textit{O. Gabber} proved that the enveloping algebra \(U\) of a finite dimensional complex solvable Lie algebra is catenary, meaning that all saturated chains of prime ideals between any two fixed primes \(P<Q\) of \(U\) have the same length. (Gabber's proof is included in [\textit{T. Levasseur} and \textit{J. T. Stafford}, Rings of differential operators on classical rings of invariants (Mem. Am. Math. Soc. 412, 1989; Zbl 0691.16019)].) The idea behind Gabber's theorem is that catenarity is a consequence of good homological and growth properties of the algebra. In the present paper the authors first state and prove an ``abstract'' version of Gabber's result, making precise the rough statement of the previous sentence; simultaneously, they derive an ``abstract'' version of Tauvel's height formula [\textit{P. Tauvel}, Bull. Soc. Math. Fr. 106, 177-205 (1978; Zbl 0399.17003)]. In the second part of the paper they show that the hypotheses required for these ``abstract'' results are satisfied by various classes of algebras associated with quantum groups.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 920025
zbMATH Keywords: enveloping algebras
zbMATH Keywords: finite dimensional complex solvable Lie algebras
zbMATH Keywords: saturated chains of prime ideals
zbMATH Keywords: catenarity
zbMATH Keywords: Tauvel's height formula
zbMATH Keywords: quantum groups
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00120-4
OpenAlex ID: W1970483208
cites work: On the field of fractions of certain quantum algebras
cites work: Q3472250
cites work: Prime spectra of quantum semisimple groups
cites work: Study of the \(q\)-commutations in the algebra \(U_ q(n^ +)\)
cites work: Freund-Witten adelic formulae for Veneziano and Virasoro-Shapiro amplitudes
cites work: Quantum Weyl algebras
cites work: Prime Factor Algebras of the Coordinate Ring of Quantum Matrices
cites work: Q3994888
cites work: Primitive ideals of \({\mathbf C}_ q[SL(3)]\)
cites work: Primitive ideals of \(\mathbb{C}_ q[SL(n)]\)
cites work: Q3717234
cites work: A simple localization of the quantized Weyl algebra
cites work: Q3677877
cites work: Q4025952
cites work: Rings of differential operators on classical rings of invariants
cites work: The quantum coordinate ring of the special linear group
cites work: Quantum groups at roots of 1
cites work: The catenarity of the positive part of the quantized enveloping algebra of a simple Lie algebra of type \(B_ 2\)
cites work: Crossed Products and Multiplicative Analogues of Weyl Algebras
cites work: Q3786586
cites work: Quantum linear groups
cites work: PBW-bases of quantum groups.
cites work: A Note on Hypercentral Group Rings
cites work: Q4010761
cites work: Homological properties of (graded) Noetherian PI rings
cites work: Sur les quotients premiers de l'algèbre enveloppante d'une algèbre de Lie résoluble
cites work: Prime ideals of quantized Weyl algebras
cites work: Spectrum of the quantized Weyl algebra
cites work: Connected graded Gorenstein algebras with enough normal elements
description: scientific article; zbMATH DE number 920025