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We consider when extensions S ⊂ R S\subset R of subalgebras of a Hopf algebra are β \beta -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H H , we show that when S ⊂ R S\subset R are Hopf algebras in the Yetter-Drinfeld category for H H , the extension is β \beta -Frobenius provided R R is finite over S S and the extension of biproducts S ⋆ H ⊂ R ⋆ H S\star H\subset R\star H is cleft. More generally we give conditions for an extension to be β \beta -Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

In this article, we attempt to generalize the result that for a commutative ring R R the outer automorphism group of R R -automorphisms of M n ( R ) {M_n}(R) is abelian of exponent n n . It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.

If G is any finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group, with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgroup of the Weyl group W, then all representations of Cw(E) are defined over Q.

Let A A be a k k -algebra graded by a finite group G G , with A 1 {A_1} the component for the identity element of G G . We consider such a grading as a “coaction” by G G , in that A A is a k [ G ] ∗ k{[G]^ \ast } -module algebra. We then study the smash product A # k [ G ] ∗ A\# k{[G]^ \ast } ; it plays a role similar to that played by the skew group ring R ∗ G R\, \ast \,G in the case of group actions, and enables us to obtain results relating the modules over A , A 1 A,\,{A_1} , and A # k [ G ] ∗ A\# k{[G]^ \ast } . After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A A and A 1 {A_1} . In particular we generalize Lorenz and Passman’s theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if π : H → H ¯ \pi :H \to \overline H is a Hopf algebra epimorphism which is split as a coalgebra map, then H H is algebra isomorphic to A # σ H A{\# _\sigma }H , a crossed product of H H with the left Hopf kernel A A of π \pi . Given any crossed product A # σ H A{\# _\sigma }H with H H (weakly) inner on A A , then A # σ H A{\# _\sigma }H is isomorphic to a twisted product A τ [ H ] {A_\tau }[H] with trivial action. Finally, if H H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A A implies that of A # σ H A{\# _\sigma }H ; in particular this is true if the (weak) action of H H is inner.

Let R R be a ring with no nilpotent elements, with extended center C C , and let E E be the set of idempotents in C C . Our first main result is that for any finite group G G acting as automorphisms of R R , there exist a finite set L ⊆ E L \subseteq E and an R G {R^G} -bimodule homomorphism τ : R → ( R L ) G \tau :R \to {(RL)^G} such that τ ( R ) \tau (R) is an essential ideal of ( R E ) G {(RE)^G} . This theorem is applied to show the following: if R R is a Noetherian, affine P I PI -algebra (with no nilpotent elements) over the commutative Noetherian ring A A , and G G is a finite group of A A -automorphisms of R R such that R G {R^G} is Noetherian, then R G {R^G} is affine over A A .

If R is an associative ring, we consider the special Jordan ring R + {R^ + } , and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of R + {R^ + } , and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.