Jeffrey Bergen

Smash products and outer derivations

LetR be a prime ring andL a Lie algebra acting onR as “Q-outer” derivations (if charR=p≠0, assume thatL is restricted). We study ideals and the center of the smash productR #U(L) (respectivelyR #u(L) ifL is restricted) and use these results to study the relationship betweenR and the ring of constantsRL. More generally, for any finite-dimensional Hopf algebraH acting onR such thatR #H satisfies the “ideal intersection property”, we useR #H to study the relationship betweenR and the invariant ringRH.

Radicals of crossed products of enveloping algebras

LetL be a Lie algebra over a fieldK which acts asK-derivations on aK-algebraR. Then this action determines a crossed productR *U(L) whereU(L) is the enveloping algebra ofL. The goal of this paper is to describe the Jacobson radical ofR * U(L) forL≠0. We are most successful whenR is a p.i. algebra or Noetherian. In more general situations we at least obtain upper and lower bounds forJ(R * U(L)) which are ideals extended fromR. Furthermore, we offer an interesting example in all characteristics of a commutativeK-algebraC which admits a derivationδ such thatC isδ-prime but not semiprime.

Irreducible actions and faithful actions of hopf algebras

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsAH, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : DH]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

Ideals and Quotients in Crossed Products of Hopf Algebras

Let H be a Hopf algebra over a field k, acting on the k-algebra R with action twisted by a cocycle (T such that the crossed product algebra R #, H can be constructed. This paper is concerned with the relationship between the ideals of R #, Hand those of R, and with computing the extended cen- ter and symmetric quotient ring of R #, H in terms of the extended center C and symmetric quotient ring Q of R. Our best results are obtained when H is an irreducible Hopf algebra, or more generally when H is of the form K # kG, where K is irreducible and kG is the group algebra of a group G. Some examples of irreducible Hopf algebras are enveloping algebras of Lie algebras U(L) along with their restricted counterparts u(L) in characteristic

SKEW DERIVATIONS WITH CENTRAL INVARIANTS

If r is an automorphism and d is a r-derivation of a ring R, then the subring of invariants is the set R(d) fl† r ‘ R r d(r) fl 0·. The main result of this paper is ‘let R be a semiprime ring with an algebraic r-derivation d such that R(d) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

Constants of Lie algebra actions

Let R be an associative algebra over a field k and let L be a Lie algebra over k acting on R as derivations. We will be interested in studying the relationship between R and the ring of constants RL. Our situation is a special case of the study of rings of invariants of Hopf algebra actions and is also analogous to another special case of Hopf algebra actions, namely, the study of fixed rings of finite group actions. Many of the results in this paper are analogs of results on group actions and give rise to more general questions regarding Hopf algebra actions. In Section 1, we study actions on non-nilpotent algebras. We first show that any algebraic derivation has non-zero constants and then prove that any finite dimensional nilpotent Lie algebra of algebraic derivations must act with non-zero constants. It then follows, as a corollary, that any finite dimensional nilpotent restricted Lie algebra must act with non-zero constants. These are analogs of results on groups proved in [ 12, 5). In Section 2, we prove a necessary condition for the existence of nonzero invariants of certain Hopf algebra actions. We then use this result to prove the converse of our theorem on nilpotent restricted Lie algebras. We conclude the section with several questions on general Hopf algebra actions.

Lie ideals with regular and nilpotent elements and a result on derivations

LetU⊄Z be a Lie ideal of a ringR. We examine those ringsR in which everyu∈U is either regular or nilpotent and prove that ifR has no non-zero nil left ideals then eitherR is a domain or an order in the 2×2 matrices over a field. We proceed by first examining ringsR with no non-zero nil left ideals possessing a derivationd≠0 such thatd (x) is nilpotent or invertible, for allx∈R. It is shown that such a ring must either be a division ring or the 2×2 matrices over a division ring. We also prove similar results for semiprime rings where the various indices of nilpotence are assumed to be bounded.

On Rings with Locally Nilpotent Skew Derivations

In this article, we examine algebras with a locally nilpotent q-skew σ-derivation d when there is an element x such that d(x) = 1 and either q is not a root of 1 or q = 1 in characteristic zero. When characteristic p > 0, we also examine the situation where d is an ordinary derivation.

Invariants of skew derivations

If σ is an automorphism and δ is a σ-derivation of a ring R, then the subring of invariants is the set R(δ) = {r ∈ R | δ(r) = 0}. The main result of this paper is Theorem. Let δ be a σ-derivation of an algebra R over a commutative ring K such that δ(r) + an−1δn+k−1(r) + · · · + a1δ(r) + a0δ(r) = 0, for all r ∈ R, where an−1, . . . , a1, a0 ∈ K and a0−1 ∈ K. (i) If Rn+1 6= 0, then R(δ) 6= 0. (ii) If L is a δ-stable left ideal of R such that l.annR(L) = 0, then L (δ) 6= 0. This theorem generalizes results on the invariants of automorphisms and derivations. If R is a ring with an automorphism σ, we say that an additive map δ : R −→ R is a σ-derivation if δ(rs) = δ(r)s + σ(r)δ(s), for all r, s ∈ R. We define the subring of invariants to be the set R = {r ∈ R | δ(r) = 0}. It was shown in [HN] that algebraic automorphisms always act with nonzero invariants on nonnilpotent algebras. The analogous result for algebraic derivations was proven in [B]. The simplest examples of σ-derivations are ordinary derivations, which occur when σ is the identity map, as well as maps of the form 1−σ. Therefore the results in this paper generalize results on the invariants of automorphisms and derivations. However, the results on automorphisms and derivations were obtained using group-graded rings, whereas our arguments are entirely combinatorial. In fact, we will present an example in which the 0-eigenspace of a σ-derivation is not a subring, thus the techniques of group-graded rings cannot be applied to this more general situation. Since we would like to apply our results to prove that various subrings and one-sided ideals contain nonzero invariants, we will not be assuming that our rings have a unit element. Received by the editors December 29, 1995 and, in revised form, July 2, 1996. 1991 Mathematics Subject Classification. Primary 16W20, 16W25, 16W55. The first author was supported by the University Research Council at DePaul University. Both authors were supported by Polish KBN Grant 2 PO3A 050 08. Much of this work was done when the first author was a visitor at the University of Warsaw, Bia lystok Division and the second author was a visitor at DePaul University. We would like to thank both universities for their hospitality. c ©1997 American Mathematical Society

Invariants of Uq(sℓ(2)) and q-skew derivations

Abstract If δ is a q-rmskew derivation of a ring R, then the subring of invariants is R (δ) − r ϵ R ¦ δ(r) = 0 . We prove Theorem. Let δ be a q-skew derivation which is algebraic in its action on the K-algebra R. If R is (σ, δ)-semiprime and I ≠ 0 is a (σ, δ)-stable ideal of R, then I(δ) is a nonnilpotent ideal of R(δ). This result is used to examine the actions of the Hopf algebra H = Uq(sl(2)). We show, under certain natural hypotheses, that for any H-stable ideal I ≠ 0 of a semiprime ring, the invariants of I under the action of Uq(sl(2)) are nonnilpotent.