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We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar's smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct., 12 pages

We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group $S_n$, namely the bismash products $H_n = k^{C_n}# kS_{n-1}$ and its dual $J_n = k^{S_{n-1}}# kC_n = (H_n)^*,$ where $k$ is an algebraically closed field of characteristic 0. Both algebras are constructed using the standard representation of $S_n$ as a factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove that for $H_n$, the indicators of all simple modules are +1. For the dual Hopf algebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values either 0 or 1. When $n = p$, a prime, we obtain a precise result as to which representations have indicator +1 and which ones have 0; in fact as $p \to \infty$, the proportion of simple modules with indicator 1 becomes arbitrarily small. We also prove a result about Frobenius-Schur indicators for more general bismash products $H =k^G# kF$, coming from any factorizable group of the form $L = FG$ such that $F\cong C_p.$ We use the definition of Frobenius-Schur indicators for Hopf algebras, as described in work of Linchenko and the second author, which extends the classical theorem of Frobenius and Schur, for a finite group $G.$

The Hopf order of an element $h$ of a Hopf algebra $H$ is the least $n$ such that the $n$-th Hopf power of $h$ is trivial. For some bismash product Hopf algebras obtained from factorizable groups (including Drinfeld doubles of some groups) we compute the dimensions of the spaces of elements with trivial $n$-th Hopf powers, and determine which Hopf orders of elements are possible. We discuss the behavior under twists and duality., 28 pages

In this paper we obtain algebraic analogs of certain known results on crossed products of von Neumann algebras and of C ∗ -algebras. Of particular interest here is the Connes spectrum, introduced by A. Connes in his classification of factors of type III. In addition, we are concerned with certain von Neumann algebra results contained in the work of Connes, Takesaki, and Nakagami and with certain, more algebraic, C ∗ -algebra results from the work of Olesen, Pederson, and Kishimoto. Let G be a finite group and let A be a G -graded ring with 1. Then there exists an extension ring of A , determined by this structure, called the smash product of A by G ∗ . This overring comes from the theory of Hopf algebras and we denote it by A # G ∗ . In the first half of this paper, we introduce and study the Connes and strong Connes subgroups of G and we relate them to the ideal structure of A # G ∗ . In particular we obtain criteria for A # G ∗ to be prime or simple. In a different direction, suppose A is a K -algebra and G is a finite abelian group of K -automorphisms of A . If the field K contains appropriate roots of unity, then the skew group ring AG has a natural smash product structure A # G ∗ where G = Hom (G,K′) is the dual group of G . The problem is to explicitly determine the Connes subgroup of G in terms of the action of G on A . This is achieved in the second half of the paper under the additional assumption that A is a prime ring. A more detailed discussion of the operator algebra sources of our results will be given in the Appendix.

We prove that if the dimension of any irreducible module for a finite-dimensional algebra over an algebraically closed field divides the dimension of the algebra, then the same is true of any crossed product of that algebra with a group algebra or its dual, provided the characteristic of the field does not divide the order of the group. Kaplansky's Conjecture regarding dimensions of irreducible modules for Hopf algebras then follows for those finite-dimensional semisimple Hopf algebras constructed by a sequence of crossed products involving group algebras and their duals. We show that any semisimple Hopf algebra of prime power dimension in characteristic 0 is of this type, so that Kaplansky's Conjecture holds for these Hopf algebras.

Let R, and R, be algebras with 1 over a common field F, with each (Rj: F) > 1, and let R = R, II R, denote the coproduct (often called the free product) of R, and R, over F. It is easy to show that R is always prime; in fact R is primitive in case at least one of (Ri: F) > 2 171. On the other hand, when (Ri: F) = 2, i= 1,2, it is clear from [ 1, p. 261 that R is prime PI but never primitive. If R, and R, are domains then we have Cohn’s result [2 i that R is again a domain. In studying certain automorphisms of a prime ring R one is led in a natural way to the normal closure of R, and we now briefly recall the meaning of this notion. If R, is the left quotient ring of R relative to the filter F of all nonzero two-sided ideals of R, the set N* of all units u of R-,such that uwlRu = R is called the set of normalizing elements [ 10, p. 5 ]. The automorphisms thus induced on R are just the X-inner automorphisms of Kharchenko [IO, p. 31. The subring RN of R generated by R and N = N* U (0) is called the normal closure of R. An important subset of N is the center C of R-, (called the extended center of R); it is known that C is a field. We now return to the coproduct R = R, II R, , where R L and R2 are arbitrary F-algebras. It is known in this situation that if at least one of (Ri: F) > 2, then C = F [8]; when both (R,: F) = 2, then C =F(tj G R [ 1 j. Now fix F-bases {xii u 1 for R, and ( ~7~) u 1 for R2 and call the various products of alternating xI(s and -yj’s basis monomials. The degree of a basis

In this paper we prove that if G is a finite group of automorphisms acting on a semiprime ring R such that R has no additive ] G j-torsion, then the skew group ring R*G is also semiprime. The result was heretofore known in such special cases as when G is finite abelian, R is Goldie, or R satisfies a polynomial identity [I]. Our technique of proof is to show first that (even in the presence of 1 G I- torsion or G possibly infinite) every nonzero ideal in R*G contains a nonzero element whose support consists of inner automorphisms, where inner is used in the sense of Kharchenko [5]. Using this we are able to show that R*G is semiprime provided that R is prime and B*Ginn is semiprime where Glnn is the subgroup of inner automorphisms and B is the algebra of the group in the quotient ring of R with respect to the filter of two-sided ideals. Next we show that indeed B*Ginn is semiprime when R has no additive 1 G j-torsion. Finally, in order to complete the proof, we make a reduction from R semiprime to R prime. Along the way we prove that if G (possibly infinite) is outer and

Let G be a locally compact abelian group acting continuously on a von Neumann algebra A via a homomorphism CI of G into Aut A. In 1973, Takesaki showed how an action oi of the dual group G could be naturally defined on the von Neumann algebra crossed product d x ~ G and proved that the double crossed product (4 x a G) x ti G was naturally isomorphic to & @ $?, where 6? is the ring of all bounded operators on L,(G). This duality theorem for group actions was extended to arbitrary locally compact groups G independently by Nakagami, Landstad, and StratilaVoiculescu-Zsido in the mid 1970s. This extension required replacing the action of G on &’ x r G by a “coaction” of G on & x a G. Indeed coactions of G are defined in general, and it is proved that (Jl”e x rr G) x d G N &Y 0 6? for any action c( of G on & and corresponding coaction oi of G on &?’ x ? G. A similar theorem is proved reversing the roles of c1 and oi. An exposition of this theorem may be found in Nakagami-Takesaki [7], where complete references to the literature may also be found. (Nakagami and Takesaki phrase things in a dual fashion to handle the topological subtleties more easily.) In [2], Cohen and Montgomery recognized that for finite groups G, a coaction of G on an F-algebra A, where F is a field, is just a G-grading of A. It was known that a G-grading of an algebra A is the same as an (FG)*-module algebra action of (FG)*, the Hopf algebra dual to FG, on A. Given these facts, they were able to prove the following duality theorems

Genetic testing for cancer susceptibility genes is increasingly being integrated into medical care. Test results help inform risks of the individual being tested as well as family members who could benefit from knowing the results. The responsibility for informing relatives of genetic test results falls on the proband, the first family member being tested. However, there are several challenges associated with sharing genetic test results within families including incomplete understanding of test results, emotional distance among family members, and poor communication skills. In this paper we describe the communication process between probands randomized to receive BRCA1/2 genetic test results in an enhanced versus a standard of care counseling session, and their first degree relatives with whom they shared results. We contacted 561 first degree relatives of probands who had undergone BRCA1/2 genetic testing to measure their level of understanding of the test results, their difficulty and distress upon hearing the results, the impact of the test results on their risk perception, and their intention to pursue genetic counseling/testing. 82.1 % of relatives correctly reported the test results of their proband. Distress upon hearing the test result was highest for those relatives whose proband received informative test results. Relatives reported a decrease in cancer risk perception after hearing the test results, regardless of the type of result. Intention to pursue counseling/testing was low, even among those relatives whose proband received informative test results. Male relatives were less likely to be informed of test results and more likely to forget hearing them. These results suggest ways to improve the communication process within families.

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