Finite-dimensional Simple Modules Research Articles

The Torsion Free Pieri Formula

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

Dimension formulas for the Lie superalgebra sl(m/n)

Although character formulas for simple finite-dimensional modules of the Lie superalgebra sl(m/n) are not known in general, they are known in the cases of so-called typical and of singly atypical modules. For singly atypical modules, however, the character formula is such that the classical limiting procedure yielding a dimension formula from a character formula does, in general, not work. In this article, another approach is presented yielding an explicit dimension formula for all singly atypical modules of sl(m/n) in terms of the highest weight labels.

Simple Cn modules with multiplicities 1 and applications

In this paper we show that there exist exactly two nonequivalent simple infinite dimensional highest weight Cn modules having the property that every weight space is one dimensional. The tensor products of these modules with any finite-dimensional simple Cn module are proven to be completely reducible and we provide an explicit decomposition for such tensor products. As an application of these decompositions, we obtain two recursion formulas for computing the multiplicities of simple finite dimensional Cn modules. These formulas involve a sum over subgroups of index 2 in the Weyl group of Cn.

Irreducible Representations of the 4-Dimensional Sklyanin Algebra at Points of Infinite Order

In 1982 Sklyanin ( Funct. Anal. Appl. 16 (1982), 27-34) defined a family of graded algebras A( E, τ), depending on an elliptic curve E and a point τ ∈ E which is not 4-torsion. Basic properties of these algebras were established in Smith and Stafford ( Compositio Math. 83 (1992), 259-289) and a study of their representation theory was begun in Levasseur and Smith ( Bull. Soc. Math. France 121 (1993), 35-90). The present paper classifies the finite dimensional simple A-modules when τ is a point of infinite order. Sklyanin ( Funct. Anal. Appl. 17 (1983), 273-284) defines for each k ∈ N a representation of A in a certain k-dimensional subspace of theta functions of order 2( k − 1). We prove that these are irreducible representations, and that any other simple module is obtained by twisting one of these by an automorphism of A. The automorphism group of A is explicitly computed. The method of proof relies on results in Levasseur and Smith. In particular, it is proved that every finite dimensional simple module is a quotient of a line module. An important part of the analysis is a determination of the 1-critical A-modules, and the fact that such a module is (equivalent to) a quotient of a line module by a shifted line module.