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Let H be a finite dimensional Hopf algebra over an algebraically closed field. We show that if H is commutative and the coradical HO is a sub Hopf algebra, then the canonical inclusion HO -+ H has a Hopf algebra retract; or equivalently, if H is cocommutative and the Jacobson radical J(H) is a Hopf ideal, then the canonical projection H -+ H/J(H) has a Hopf algebra section. For a Hopf algebra H we denote the coradical (i.e. the sum of the simple subcoalgebras of H) by Ho, and the Jacobson radical by J(H). If 7: H -> K is a surjective (resp. injective) Hopf algebra map we say it splits if there exists a Hopf algebra map T: K -* H with 77 ? T = IK (resp. T ? 77 = IH). The purpose of this paper is to prove that if H is a finite dimensional Hopf algebra over an algebraically closed field we have the following: (A) If H is commutative and Ho is a sub Hopf algebra, then the canonical inclusion Ho -H splits as a map of Hopf algebras; or equivalently, (B) If H is cocommutative and J(H) is a Hopf ideal, then the canonical projection H -H/J(H) splits as a map of Hopf algebras. If follows from the results of [3] that the existence of a Hopf algebra splitting in (A) or (B) induces a semidirect product decomposition of the Hopf algebra H, and that such splittings are necessarily unique. For the standard facts about Hopf algebras see [1] or [7]; for splittings and exact sequences see [3]. It is easy to see that (A) and (B) are equivalent, for by finite dimensionality we have J(H*) = (H0)' and so Ho _ (H*/J(H*))*. Thus a splitting in one case induces a splitting in the other by transposing. We shall verify (B). We begin by establishing a special case of (B) which is valid over any field. If G is a group, let k[G] denote the group algebra of G over k. PROPOSITION 1. Let H = k[G] where G is a finite group and k is any field. If J(H) is a Hopf ideal of H then the canonical projection g: H -H/J(H) splits as a map of Hopf algebras. PROOF. If the characteristic of k is zero (or is relatively prime to the order Received by the editors January 20, 1976. AMS (MOS) subject classifications (1970). Primary 16A24; Secondary 13E10.

A semidirect product decomposition for certain Hopf algebras over an algebraically closed field