On polynomial representations of strange Lie superalgebras of <Emphasis Type="Italic">Q</Emphasis>-type

In this paper, both canonical and noncanonical polynomial representations of Lie superalgebara of Q-type are investigated. It turns out that not all these representations are completely reducible. Moreover, the representation spaces has only two proper submodules when it is completely reducible, and has a unique composition series when it is not completely reducible.     

Left and right negatively orderable semigroups where every element has a left and a right identity

We study two classes of modules for order preserving symplectic and order preserving even special orthogonal rook monoids. The first class of modules are indexed by admissible sets, and the second are the regular representations. For the first class we show that every submodule is cyclic and completely reducible, and for each irreducible component when restricted to a module over a proper submonoid we explore its module properties. For the second we provide explicitly their decompositions into irreducible submodules using admissible sets.     

局限李超代数的局限表示

本文对如下结果给出一个构造性、初等的、简洁的证明:有限维局限李超代数的有限维局限表示均完全可约当且仅当它的奇部为0而偶部是一个环面.这使得李代数中相应结果成为特例.     

Algebraic Representation Theory of Partial Algebras

. This paper develops an algebraic representation theory of partial algebras. We introduce the notion of irreducibility for representations of a partial algebra, prove necessary and sufficient conditions for a representation to be (a) irreducible or (b) completely reducible, and develop aspects of the decomposition theory of completely reducible representations. An application to boson quantum field theory is furnished.     

Representations and Cocycle Twists of Color Lie Algebras

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra .Presented by A. Verschoren.     

-QUASIALGEBRAS

Abstract

The purpose of this work is to construct completely reducible vertex representations for the toroidal Lie algebra of type F ₄.     

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

Some evaluation of q-analogues of Euler sums

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several classes of q-Euler sums in terms of q-polylogarithms and q-special functions. Furthermore, we develop new closed form representations of sums of quadratic and cubic parametric q-Euler sums. Finally, we can find that the q-Euler sums are reducible to the classical Euler sums when q approaches 1.     

Properties and examples of FCR-algebras

An algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C ^*-algebras and to localizations. Moreover, we show that “small” products and sums of FCR-algebras are again FCR. Received: 25 October 2000     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Representations of semigroups of partial isometries

We present a new proof that the irreducible representationsof the von Neumann algebra generated by a strongly continuoussemigroup of partial isometries of index 1 are unique up toequivalence, as well as a proof that when such an algebra isa factor, its representations are completely reducible. As anapplication, we show that the irreducible representations ofa strongly continuous semigroup of isometries {U(), 0} suchthat are equivalent.     

Geometry of the Mathieu groups and Golay codes

A brief review is given of the linear fractional subgroups of the Mathieu groups. The main part of the paper then deals with the projective interpretation of the Golay codes; these codes are shown to describe Coxeter’s configuration inPG(5,3) and Todd’s configuration inPG(11,2) when interpreted projectively. We obtain two twelve-dimensional representations ofM ₂₄. One is obtained as the collineation group that permutes the twelve special points inPG(11,2); the other arises by interpreting geometrically the automorphism group of the binary Golay code. Both representations are reducible to eleven-dimensional representations ofM ₂₄.     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

交换矩阵半群的可约性

首先分别给出单生矩阵半群或者摹群不可约、不可分解以及完全可约的充分必要条件,其次讨论一般域上矩阵半群的可约性的一些条件,最后特别地讨论实数域上矩阵半群的可约性,完全确定了实数域上对称和反对称矩阵组成的不可约交换矩阵半群.     

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.

     

Possible splitting of energy levels of atoms or ions in octahedral crystals due to sets of equivalent neighbouring defects

Divalent ions when doped in alkali halide single crystals will go in alkali sites and form associate pairs with neighbouring cation vacancies or other impurity defects. As there will be sets of equivalent positions for such associated pairs, their energy levels will be degenerate. The reducible representations of these associated pairs involving first to sixth nearest neighbours have been reduced to the corresponding irreducible representations of the point group O _h , for cases having rock-salt structure as well as for cases having cesium chloride structure. It is pointed out that if there is a rapid hopping of the vacancies or the defects in the crystal one may expect because of the possible interactions in the system, a splitting of the energy levels of the pairs as indicated by the different irreducible representations. It is stated that this kind of splitting may be expected also in systems in which associated magnetic pairs are formed by two neighbouring spins.     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

Degenerate series representations of the universal covering group of SU(2, 2)

We prove a reducibility criterion for certain families of representations induced from irreducible finite dimensional representations of the 11-dimensional parabolic subgroup of the universal covering group of SU(2, 2). If an induced representation is reducible and can be considered as a representation of SU(2, 2) as well, we compute the number of composition factors.