Invariants of \(U_q(\text{sl}(2))\) and \(q\)-skew derivations
type: Item
instance of: scholarly article
title: Invariants of \(U_q(\text{sl}(2))\) and \(q\)-skew derivations
zbMATH Open document ID: 0934.16031
DOI: 10.1016/S0022-4049(97)00179-5
author: Jeffrey Bergen
author: Piotr Grzeszczuk
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-03T00:00:00Z
review text: Let \(R\) be an algebra over a field \(K\). A \(K\)-linear map \(\delta\colon R\to R\) is called a \(q\)-skew derivation if there exist a nonzero \(q\in K\) and a \(K\)-linear automorphism \(\sigma\) of \(R\) such that for all \(r,s\in R\), \(\delta(rs)=\delta(r)s+\sigma(r)\delta(s)\), and \(\delta\sigma=q\sigma\delta\). A subset \(A\subseteq R\) with the properties \(\sigma(A)=A\) and \(\delta(A)\subseteq A\) is called \((\sigma,\delta)\)-stable. If \(A\) is a \((\sigma,\delta)\)-stable subring of \(R\), let \(A^{(\delta)}=\{a\in A\mid\delta(a)=0\}\) denote the invariants. An algebra \(R\) is said to be \((\sigma,\delta)\)-semiprime if it has no nonzero nilpotent \((\sigma,\delta)\)-stable ideals. The authors prove the following Theorem. Let \(\delta\) be a \(q\)-skew derivation which is algebraic in its action on the \(K\)-algebra \(R\). If \(R\) is \((\sigma,\delta)\)-semiprime and \(I\neq 0\) is a \((\sigma,\delta)\)-stable ideal of \(R\), then \(I^{(\delta)}\) is a nonnilpotent ideal of \(R^{(\delta)}\). The authors use this result to study actions of the Hopf algebra \(U_q(\text{sl}(2))\), and show, under some natural assumptions, that for any nonzero \(H\)-stable ideal \(I\) of a semiprime ring, the invariants of \(I\) under the action of \(U_q(\text{sl}(2))\) are nonnilpotent.
reviewed by: Tomasz Brzeziński
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1308275
zbMATH Keywords: skew derivations
zbMATH Keywords: Hopf algebras
zbMATH Keywords: invariants
zbMATH Keywords: algebraic actions
zbMATH Keywords: semiprime rings
MaRDI profile type: Publication
cites work: Invariants of skew derivations
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cites work: Constants of algebric derivations
cites work: Q4744461
cites work: Biinvertible actions of Hopf algebras
description: scientific article; zbMATH DE number 1308275