Daniel Britten

On modules of bounded multiplicities for the symplectic algebras

Simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product. Also, it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible.

Embedding rings in completed graded rings 3. Algebras over general k

Abstract It is shown that any associative algebra R over a commutative ring k such that R n + 1 = {0} can be embedded in a graded k -algebra ⊕ H i , such that H i ≠ {0} only for 1 ⩽ i ⩽ 2 n − 1. By the results of the first paper in this series, R can therefore be embedded in (2 n − 1 + 1) × (2 n − 1 + 1) strictly upper triangular matrices over a commutative k -algebra. For k any integral domain not a field, this is in fact a best result, in contrast with the case of k a field, where one can replace “2 n − 1 ” by “ n ” (see Part 2 of this series). More generally, if R is a nonunital k -algebra with a strictly positive-integer-valued filtration function, then R can be mapped into a completed Z -graded k -algebra S by a homomorphism f such that v ( a ) ⩽ v ( F ( a )) ⩽ 2 v ( a ) − 1 , where v denotes the given filtration function on R and the grading-filtration on S . If all values of the filtration function on R are powers of 2, one can even get v ( f ( a )) = v ( a ).

Submodule Lattice of Generalized Verma Modules

Abstract Let 𝒢 be a simple finite dimensional Lie algebra over the complex numbers and let 𝒢¯ = 𝒢1 ⊕…⊕ 𝒢 k be a regular semisimple subalgebra of 𝒢 with each 𝒢 i being a simple algebra of type A or C. It is shown that the lattice of submodules of a generalized Verma 𝒢-module constructed by parabolic induction starting from a simple torsion free 𝒢¯-module is almost always isomorphic to the lattice of submodules of an associated module formed as a quotient of a classical Verma module by a sum of Verma submodules. In particular, it is shown that the Mathieu admissible Verma modules involved have maximal submodules which are the sum of Verma modules.

Goldie-like conditions on Jordan matrix rings

In this paper Goldie-like conditions are put on a Jordan matrix ring J = H(Rn, ya) which are necessary and sufficient for R to be a *-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.

A Structure Theorem for Rings Supporting a Discrete Fourier Transform

In this paper we show that finite commutative rings with unity generated by an invertible element can be characterized as certain direct sums of Galois rings. This result is used to analyze the structure of rings which support discrete Fourier transforms.

Tensor Products of Torsion Free C n -Modules of Finite Degree and Finite Dimensional Modules

It is known that every torsion free C n -module of finite degree is completely reducible. In this article, we provide a formula for the decomposition of the tensor product of a simple torsion free C n -module of finite degree with a simple finite dimensional C n -module. As a byproduct we obtain a recursion formula for the decomposition of the tensor product of any two simple finite dimensional C n -modules.

A constraint on the existence of simple torsion free Lie modules

For any simple Lie algebra L with Cartan subalgebra H the classification of all simple H-diagonalizable L-modules having a finite-dimensional weight space is known to depend on determining the simple torsion-free Lmodules of finite degree. It is further known that the only simple Lie algebras which admit simple torsion-free modules of finite degree are those of types An and Cn . For the case of An we show that there are no simple torsion-free Anmodules of degree k for n > 4 and 2 1, and C,, n > 2, possess torsion-free modules. It is easy to see that no simple torsion-free modules of minimal weight space dimension greater than 1 exist for A1 . The existence of simple torsion-free An-modules of arbitrary degree for n = 2, 3 has been established in [BBL]. In this note, we further restrict the existence of torsion-free A,-modules by proving Main Theorem. There are no simple torsion-free A,-modules of degree k for n>4 and 2< k<n2. Received by the editors October 30, 1991 and, in revised form, November 11, 1993. 1991 Mathematics Subject Classification. Primary 1 7B 10. The first author was supported in part by NSERC Grant #A8471. The second author was supported in part by NSERC Grant #A7742. @ 1995 American Mathematical Society