Simple modules of the Witten-Woronowicz algebra.
type: Item
instance of: scholarly article
title: Simple modules of the Witten-Woronowicz algebra.
zbMATH Open document ID: 1047.16001
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Let \(k\) be an algebraically closed field, \(D=k[H,C]\) a polynomial algebra, \(\xi\in k^*\) not a root of 1 and \(a=a(H,C)\in D\). A generalized Weyl algebra in the sense of V. V. Bavula is the associative \(k\)-algebra \(A\) generated by the elements \(X,Y,H,C\) subject to the defining relations \[ XH=\xi HX,\;YH=\xi^{-1}HY,\;XC=\xi^p CX,\;YC=\xi^{-p}CY,\;YX=a(H,C),\;XY=a(\xi H,\xi^pC), \] where \(p\) is a positive integer. It is easy to check that the elements \(H,C,H^p-\lambda\) are normal in \(A\) for any \(\lambda\in k\). The main result of the paper shows that a simple \(A\)-module is in fact a module over one of the algebras \[ A/(H),\quad A/(C),\quad A/(H^p-\lambda),\qquad\lambda\in k^*. \] It is mentioned that Witten's second deformation and Woronowicz's deformation are isomorphic and they belong to the class of generalized Weyl algebras.
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S80
zbMATH DE Number: 2037113
zbMATH Keywords: simple modules
zbMATH Keywords: skew polynomial extensions
zbMATH Keywords: generalized Weyl algebras
zbMATH Keywords: deformations
author: V. V. Bavula
author: Fred Van Oystaeyen
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.06.013
OpenAlex ID: W2017518875
cites work: Finite dimensionality of \(\text{Ext}^ n\)'s and \(\text{Tor}_ n\)'s of simple modules over one class of algebras
cites work: Q4021409
cites work: Q4288679
cites work: Q4888449
cites work: Isomorphism problems and groups of automorphisms for generalized Weyl algebras
cites work: Generalized Weyl algebras are tensor Krull minimal
cites work: The simple modules of certain generalized crossed products
cites work: Q4227233
cites work: Down-up algebras
cites work: Noncommutative deformations of type-\(A\) Kleinian singularities
cites work: Primitivity in skew Laurent polynomial rings and related rings
cites work: Noetherian down-up algebras
cites work: Down-up algebras and their representations.
cites work: Q4850053
cites work: Q4010761
cites work: Q4693955
DOI: 10.1016/J.JALGEBRA.2003.06.013
description: scientific article; zbMATH DE number 2037113