Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre’s notion of G-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index (H : H◦) is...

It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is semisimple and each irreducible admissible V -module is ordinary. A contravariant form on a Verma type admissible V -module is constructed and the radical is exactly the maximal proper submodule. As an application the rationality of V + L for any positive definite even lattice is obtained.

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of with $A \cong B$, nonsingular direct summand $M$, then $M$. Various examples $C_{21}$-modules presented. Some basic properties these modules investigated. It shown that the class rings $R$ over which every $C_2$-module exactly right SI-rings. Also, we prove for ring $R$, $R$-module has $(C_{21})$ if only t-semis...

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p > 2h − 2 and a given tilting module has h...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

Let L be a free Lie algebra of finite rank r over a field . For each positive integer n, denote the degree n homogeneous component of L by L. The group of graded algebra automorphisms of L may be identified with GLr; ‘ in such a way that L1 becomes the natural module, and then the L are referred to as the Lie powers of this module. Understanding the GLr; ‘-module structure of the L may be tho...

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras. The classical First and Second Whitehead Lemmata state that the first, respectively second, cohomology group of a finite-dimensional semisimpl...

This note is an outline of some of the author's recent work on the relative theory of Tamagawa numbers of semisimple algebraic groups. The details and applications will be published elsewhere. Let k be an algebraic number field of finite degree over Q, let G be a connected semisimple algebraic group defined over k. G admits one and only one simply connected covering (G, ir) defined over k (exce...

The existence and construction of self-dual codes in a permutation module of a finite group for the semisimple case are described from two aspects, one is from the point of view of the composition factors which are self-dual modules, the other one is from the point of view of the Galois group of the coefficient field.