Central extensions of three dimensional Artin-Schelter regular algebras
type: Item
instance of: scholarly article
title: Central extensions of three dimensional Artin-Schelter regular algebras
zbMATH Open document ID: 0876.17019
DOI: 10.1007/PL00004532
author: Michel Van den Bergh
author: Lieven Le Bruyn
author: S. Paul Smith
published in: Mathematische Zeitschrift
publication date: 1997-11-13T00:00:00Z
full work available at URL: https://eudml.org/doc/174881
review text: Artin-Schreier regular (AS-regular, for short) algebras of dimension \(r\) are certain graded algebras having very good homological properties and the same Hilbert series as a polynomial ring in \(r\) variables. The classification of these algebras for \(r=3\) was undertaken in \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)], \textit{M. Artin, J. Tate} and \textit{M. van den Bergh} [The Grothendieck Festschrift Vol. 1, Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)]. For general \(r\), there is an important family of examples -- the so called Sklyanin algebras [see e. g. \textit{J. Tate} and \textit{M. van den Bergh}, Invent. Math. 124, 619-647 (1996; Zbl 0876.17011)]. In the present paper, another class of 4-dimensional AS-regular algebras is investigated: central extensions of 3-dimensional ones. For this, the authors first consider quadratic algebras \(D\) with a normal element \(z\) of degree 1 such that \(D/\langle z \rangle\) is Koszul, and obtain precise conditions for \(D\) to be also Koszul. Then they classify central extensions of 3-dimensional AS-regular algebras which are also AS-regular algebras, and determine their point and line modules.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 975463
zbMATH Keywords: Sklyanin algebras
zbMATH Keywords: elliptic curve
zbMATH Keywords: Artin-Schreier regular algebras
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
cites work: Graded algebras of global dimension 3
cites work: Q3972720
cites work: Modules over regular algebras of dimension 3
cites work: Zero divisors in enveloping algebras of graded Lie algebras
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: Some properties of non-commutative regular graded rings
cites work: Modules over the 4-dimensional Sklyanin algebra
cites work: On Quantum Spaces of Lie Algebras
cites work: Q4006650
cites work: Q4205604
cites work: The Poincaré series of a local ring
cites work: Q4012684
cites work: Irreducible representations of the \(4\)-dimensional Sklyanin algebra at points of infinite order
cites work: Filtered Algebras and Representations of Lie Algebras
cites work: Nets of conics
OpenAlex ID: W4245784322
Recommended article: Central extensions of 3-dimensional Zinbiel algebras
Recommended article: Artin-Schelter regular algebras of global dimension three
Recommended article: Nontrivial central extensions of 3-Lie algebras
Recommended article: Ideal classes of three dimensional Artin-Schelter regular algebras.
Recommended article: Q4322752
Recommended article: Normalizing extensions of the two-Veronese of a three dimensional Artin-Schelter regular algebra on two generators
Recommended article: ON THE CENTER OF THE PROJ OFATHREE DIMENSIONAL REGULAR ALGEBRA
Recommended article: Q5224598
Recommended article: Artin–Schelter regular algebras of dimension five with three generators
Recommended article: Homologies of cubic Artin-Schelter regular algebras
description: scientific article