Coactions of E(n) on Clifford Algebras

We obtain a characterization of coactions of Sweedler's Hopf algebra on a finite-dimensional algebra A. We show that each coaction is completely determined by the choice of an involution and of a suitable skew-derivation of A. We classify all involutions and skew-derivations of A when A is a four-dimensional Clifford algebra and use this complete classification to determine which coactions allow us to construct rt(h-)separable and Frobenius cowreaths. We extend the correspondence between coactions and pairs of involutions and skew-derivations found in the four-dimensional case to algebras of higher dimension. We specify to the case A is a simple algebra and show that under this hypothesis the family of coactions on A is determined by the choice of a tuple of elements whose squares are in the center of A. Finally, assuming A is non-semisimple, we obtain a refinement of our classification of coactions in the four dimensional case by identifying isomorphic coactions.