Invariants of \(U_q(\text{sl}(2))\) and \(q\)-skew derivations
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.
scientific article; zbMATH DE number 1308275