Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension
type: Item
instance of: scholarly article
title: Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension
zbMATH Open document ID: 0892.16022
published in: Journal of Algebra
publication date: 1998-07-22T00:00:00Z
review text: This substantial paper shows that a Noetherian Hopf algebra of finite global dimension and satisfying a polynomial identity possesses attractive homological properties. Let \(H\) be an FBN Hopf algebra over a field \(k\) such that all irreducible \(H\)-modules are finite dimensional over \(k\). If \(H\) has finite global dimension \(n\) then \(H\) is an Auslander-regular Macaulay ring of Krull dimension \(n\). Moreover, \(H\) is a semiprime ring and, in case \(H\) is PI, a finite direct sum of prime rings. In particular, if \(\mathfrak g\) is a Lie algebra of finite dimension \(n\) over a field of positive characteristic then the enveloping algebra \(U({\mathfrak g})\) is Auslander regular and Macaulay of Krull and global dimension \(n\). On the other hand, if \(H\) has finite injective dimension \(n\) then \(H\) is Artin-Schelter Gorenstein and, in case \(H\) is PI, an Auslander-Gorenstein Macaulay ring of Krull dimension \(n\). These results are applied to quantized enveloping algebras \(U_q({\mathfrak g})\) and to quantized function algebras \({\mathcal O}_q[G]\), both in the case of a generic parameter \(q\) and in the case where \(q\) specializes to a primitive \(l\)th root of unity \(\varepsilon\). Here \(\mathfrak g\) is a finite dimensional complex semisimple Lie algebra and \(G\) is the connected simply connected semisimple Lie group with Lie algebra \(\mathfrak g\). As usual, \(l\) is odd and prime to 3 if \(\mathfrak g\) involves a factor of type \(G_2\). The second major theme in the paper is motivated by the following question: let \(\Lambda\) be a prime Noetherian ring finitely generated as a module over its centre \(Z\) where \(Z\) is an affine algebra over an algebraically closed field \(k\); for which \(m\in\max Z\) is \(\Lambda_m\) an Azumaya algebra over \(Z_m\)? The set of such \(m\) in \(Z\) is denoted by \({\mathcal A}_\Lambda\) (the Azumaya locus) and the singular locus is given by \({\mathcal S}_\Lambda=\{m\in Z:Z_m\) is not regular\}. The au…
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16R99
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1106639
zbMATH Keywords: FBN rings
zbMATH Keywords: PI rings
zbMATH Keywords: global dimension
zbMATH Keywords: polynomial identities
zbMATH Keywords: FBN Hopf algebras
zbMATH Keywords: irreducible modules
zbMATH Keywords: Auslander-regular Macaulay rings
zbMATH Keywords: Krull dimension
zbMATH Keywords: semiprime rings
zbMATH Keywords: finite direct sums of prime rings
zbMATH Keywords: injective dimension
zbMATH Keywords: Auslander-Gorenstein Macaulay rings
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: quantized function algebras
zbMATH Keywords: finite dimensional complex semisimple Lie algebras
zbMATH Keywords: prime Noetherian rings
zbMATH Keywords: affine algebras
zbMATH Keywords: Azumaya algebras
author: Kenneth A. Brown
author: Kenneth R. Goodearl
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
OpenAlex ID: W2057014450
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Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension (DOI)
DOI: 10.1006/JABR.1997.7109

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