Abstract. Sweedler's Hopf algebra H4 plays a leading role in the study of
Hopf algebras. This is shown by the great amount of generalizations present
in literature, among which are Radford's algebras Am,n, Taft's algebras Hn,q,
and a family of pointed Hopf algebras, whose coradical is the group algebra
kC2, denoted E(n) - introduced by [BDG] and investigated by [CD]. Following
the steps of [CY], we are able to characterize the actions of H4 = E(1) on a
finite dimensional algebra A in terms of involutions and skew-derivations on
A. We also provide a dual characterization for coactions. We then specialize
to the case when A = Cl(α, β, γ), a Clifford-type algebra (see [PVO2]). In this
case we present some results on the structure of Inv(A) (the set of involutions
of A) and on the family of skew-derivations associated to each φ ∈ Inv(A).
By using this classification, we are able to explicitly determine all coactions
of H4 on the 4-dimensional Clfford algebra A. This, in turn, yields some
results about the Frobenius property and the (h-)separability of the cowreath
(A⊗Hop,H, ψ) (This is a work in progress with C. Menini and B. Torrecillas).
[BDG] M. Beattie, S. D sc lescu, L. Grünenfelder, Constructing Pointed Hopf Algebras by Ore
Extensions, Journal of Algebra, Volume 225, Issue 2 (2000), 743-770.
[CD] S. Caenepeel, S. D sc lescu, On pointed Hopf algebras of dimension 2n, Bull. London.Math.
Soc. 31 (1999), 17-24.
[CY] L. Centrone, F. Yasumura, Actions of Taft's algebras on finite dimensional algebras, Journal
of Algebra, Volume 560 (2020), 725-744.
[PVO2] F. Panaite, F. Van Oystaeyen, Cliord-type algebras as cleft extensions for some pointed
Hopf algebras, Communications in Algebra, 28:2 (2000), 585-600.