Crossed products of restricted enveloping algebras

Let K be a eld of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials. Crossed products of group algebras have played an important part in the study of group algebras and Galois theory of rings, while considerable use has also been made of the analogous constructions for universal enveloping algebras of Lie algebras. The more general notion of crossed product of a Hopf algebra, which subsumes the above cases, was introduced in full generality in 3] and independently in 2]. The object of this paper is to clarify the special case in which the Hopf algebra in question is the restricted enveloping algebra of a restricted Lie algebra, as was done for ordinary enveloping algebras in 5]. We show that three constructions so far given in the literature deene the same object up to isomorphism. In order to establish this it is necessary to compute explicit formulae relating the parameters which innuence the twisting of the p-mapping. Specializing to the case where the coeecient ring is commutative we obtain a formula which deserves further clariication.