A Semidirect Product Decomposition for Certain Hopf Algebras Over an Algebraically Closed Field

Richard K. Molnar
1976 Proceedings of the American Mathematical Society  
Let H be a finite dimensional Hopf algebra over an algebraically closed field. We show that if H is commutative and the coradical H0 is a sub Hopf algebra, then the canonical inclusion H0 -> H has a Hopf algebra retract; or equivalently, if H is cocommutative and the Jacobson radical JiH) is a Hopf ideal, then the canonical projection H -» H/J(H)ras a Hopf algebra section. For a Hopf algebra H we denote the coradical (i.e. the sum of the simple subcoalgebras of H) by H0, and the Jacobson
more » ... the Jacobson radical by J{H). If tr: 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 -tr ° t = IK (resp. t ° it = 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 H0 is a sub Hopf algebra, then the canonical inclusion //0 -» 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/JiH) 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 JiH*) = (7/0)x and so H0 ss (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.
doi:10.2307/2042031 fatcat:cimxhv2kmvbppbenbcvzr2yday