Representations of finite group schemes and morphisms of projective varieties

Given a finite group scheme \cG over an algebraically closed field $k$ of characteristic \Char(k)=p>0, we introduce new invariants for a \cG-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$to$M$that take values in Grassmannians of$M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant$j$-rank have a well-defined degree ranging between$0$and$j\rk^j(M)$, where$\rk^j(M)$is the generic$j$-rank of$M$. The extreme values are attained when the module$M$has the equal images property or the equal kernels property. We establish a formula linking the$j$-degrees of$M$and its dual$M^\ast$. For a self-dual module$M$of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back$\alpha^\ast(M)$of$M$along a$p$-point$\alpha : k[X]/(X^p) \lra k\cG$

Arbitration and the Batson Principle

As disputants more frequently utilize arbitration to resolve disputes, the likelihood that discriminatory arbitrator selection will occur also increases. While some disputants might consent to selecting an arbitrator for particular reasons, it is troublesome to think that repeat players, such as employers and businesses, might use their greater knowledge and experience with the arbitral process to gain control over the arbitrator selection process through the use of peremptory challenges. Opponents of arbitration have attempted to adopt existing legal arguments to address this problem. Unfortunately, however, neither the state action doctrine nor the use of the existing public policy exception to the enforcement of an arbitration agreement or arbitral award will be successful as a means to challenge the use of discriminatory peremptory strikes. Because existing legal arguments fail to address this growing problem, this Article proposes an amendment to the Federal and Uniform Arbitration Acts to address the problem of discriminatory arbitrator selection. The proposed statute, which would ban discrimination in the selection of an arbitrator on the basis of race, ethnicity, national origin, sex, religion, or sexual orientation, mirrors the classifications that the Batson principle encompasses and thus is justifiable for both practical and constitutional reasons

Gradings of non-graded Hamiltonian Lie algebras

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2\colon\n;\omega_2) (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2\colon\n;\omega_2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).Comment: 36 pages, to be published in J. Austral. Math. Soc. Ser.

Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and corrections in Section 4.