Skew derivations andU q (sl(2))

This note first describes the basic properties of the skew derivations on the polynomial ringkX]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U _q(sl(2)), has a unique action on CX], where “action” means that CX] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU _q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it. Both authors were supported by the NSF, S. Montgomery by grant DMS 87-00641, and S. P. Smith by DMS 87-02447.