Skew derivations and \(U_ q(sl(2))\)
After recalling the idea of a skew derivation \(\delta\) of an algebra (over a field k) with respect to an automorphism \(\sigma\) of A, the authors recall how the skew group ring \(H=k[Y]*<\sigma >\), where \(\sigma (Y)=\alpha Y\) for \(\alpha\neq 0\) in k, can be given the structure of a Hopf algebra (non-commutative, non-cocommutative), H acts (as Hopf algebra) on \(A=k[X]\), \(\sigma\) acting as automorphism and Y as \(\sigma\)- derivation. This example is then ``expanded'' to (non-commutative) polynomials in 2 variables. Thus \(A=k<x,y>/(xy-\gamma yx)\), \(\gamma\neq 0\) in k. An automorphism \(\sigma\) of A is given, as are two skew-derivations \(\partial_ 1\), \(\partial_ 2\) and \(H=k[\sigma,\sigma^{- 1},\partial_ 1,\partial_ 2]\) is given the structure of a Hopf algebra. This algebra H is then shown to be isomorphic to a subalgebra \(W_{\nu}\) of the quantum group \(U_ q(sl(2))\) for \(\nu =q^ 2\). \(W_{\nu}\) was first constructed by \textit{S. Woronowicz} [Publ. Res. Inst. Math. Sci. 23, 117-181 (1987; Zbl 0676.46050)]. However, \(W_{q^ 2}\) is not a Hopf algebra, so the authors consider \(W_ q\) instead. They determine the (essentially) two ways that C[X] is a \(W_ q\)-module algebra, and then obtain the analogous actions for \(U_ q(sl(2))\) on k[X]. The authors conclude by studying a Hopf module action of \(W_ q\) on \(A={\mathbb{C}}[x,y]/(xy-q^ 2yx)\). The homogeneous components of A are simple \(W_ q\)-modules, which remain simple as \(W_{q^ 2}\)-modules, and A is also a \(U_ q(sl(2))\)-module algebra.
scientific article; zbMATH DE number 4181569
scientific article