Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

Midcycle administration of a progesterone synthesis inhibitor prevents ovulation in primates.

Progesterone receptors appear in granuloma cells of preovulatory follicles after the midcycle gonadotropin surge, suggesting important local actions of progesterone during ovulation in primates. Steroid reduction and replacement during the gonadotropin surge in macaques was used to evaluate the role of progesterone in the ovulatory process. Animals received gonadotropins to induce development of multiple preovulatory follicles, followed by human chorionic gonadotropin (hCG) administration (day 0) to promote oocyte (nuclear) maturation, ovulation, and follicular luteinization. On days 0-2, animals received no further treatment; a steroid synthesis inhibitor, trilostane (TRL); TRL + R5020; or TRL + dihydrotestosterone propionate (DHT). On day 3, ovulation was confirmed by counting ovulation sites and collecting oviductal oocytes. The meiotic status of oviductal and remaining follicular oocytes was evaluated. Peak serum estradiol levels, the total number of large follicles, and baseline serum progesterone levels at the time of hCG administration were similar in all animals. Ovulation sites and oviductal oocytes were routinely observed in controls. Ovulation was abolished in TRL. Progestin, but not androgen, replacement restored ovulation. Relative to controls, progesterone production was impaired for the first 6 days post-hCG in TRL, TRL + R5020, and TRL + DHT. Thereafter, progesterone remained low in TRL but recovered to control levels with progestin and androgen replacement. Similar percentages of mature (metaphase II) oocytes were collected among groups. Thus, steroid reduction during the gonadotropin surge inhibited ovulation and luteinization, but not reinitiation of oocyte meiotic maturation, in the primate follicle. The data are consistent with a local receptor-mediated role for progesterone in the ovulatory process.