Differential operators on quantum spaces for \(GL_ q(n)\) and \(SO_ q(n)\)
This paper is concerned with rings of \(q\)-differential operators \(\text{Diff}_ G(n)\) (as defined by Wess and Zumino) on the `quantum planes' associated with the quantum groups \(G=GL_ q(n)\) and \(G=SO_ q(n)\). In both of these cases the author explicitly constructs normal elements of \(\text{Diff}_ G(n)\) and is thus able to show that a suitable extension ring of \(\text{Diff}_ G(n)\) contains a tensor product of \(n\) copies of \(\text{Diff}_ \alpha(1)\), for suitable \(\alpha\). Here \(\text{Diff}_ \alpha(1)=\mathbb{C}[x][\partial]\) where \(\partial\) is the \(\alpha\)-derivation with \(\partial(x)=1\), so that \(\partial x-\alpha x\partial=1\). The author interprets this as an `isomorphism' between the classical and quantized rings of differential operators for generic values of the parameter \(q\). D. A. Jordan has pointed out that, in the \(GL_ q(n)\) case, replacing \(q^ 2\) by \(q\) in the relations allows one to obtain the required tensor product simply by inverting normal elements.
scientific article; zbMATH DE number 89336