仿射李代数的水平零的Whittaker模

对任意的仿射李代数■,作者构造了一类水平为零的imaginary Whittaker ■模.同时证明了这类模在某些给定条件下是单的.     

Whittaker Categories and Whittaker Modules for Lie Superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, the construction of Whittaker modules from Whittaker modules for the even part.     

Whittaker Modules for the Insertion-Elimination Lie Algebra

This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism.     

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n₁ and n₂ from our list, that are positive parts of the affine Kac–Moody algebras A₁⁽¹⁾ and A₂⁽²⁾, respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

     

Verma-Type Modules for Quantum Affine Lie Algebras

Let be an untwisted affine Kac–Moody algebra and M_J() a Verma-type module for with J-highest weight P. We construct quantum Verma-type modules M_J^q() over the quantum group , investigate their properties and show that M_J^q() is a true quantum deformation of M_J() in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules. Presented by A. VerschorenMathematics Subject Classifications (2000) 17B37, 17B67, 81R50.The first author is a Regular Associate of the ICTP. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.     

Graded Multiple Analogs of Lie Algebras

Graded analogs of (n,k,r)-Lie algebras (in particular, of Nambu–Lie algebras), introduced by the authors, are defined and their general property are studied.     

Graded Lie Algebras of Maximal Class

We study graded Lie algebras of maximal class over a field of positive characteristic . A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct pairwise non-isomorphic such algebras, and soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.

     

Graded Lie Algebras and Dynamical Systems

We define the graded Lie algebras generated by ergodic transformation with invariant measure. This algebra is the central extension of Lie algebras which is associated with the usual crossed product. At the same time, it is the special case of algebras with continuum root systems which were defined by the author and M. Saveliev at the beginning of Nineties. Examples of systems with a discrete spectrum are considered.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

Capability of Crossed Modules of Lie Algebras

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module.     

Simple Modules for the Restricted Cartan Type Lie Algebras

The simple modules with homogeneous characters are considered, their dimension formulas are determined.Presented by D. Passman.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

Maximal Graded Subalgebras of Loop Toroidal Lie Algebras

Let be a central simple Lie algebra over a field . We study the maximal ℤⁿ-graded subalgebra of . Mathematics Subject Classifications (2000) 17B70, 17B65, 17B67.     

某些$W$-型李代数的Verma模

本文我们研究了两类$W$-型李代数$g(\lambda)$的Verma模的结构. 在一定条件下,我们决定了这些Verma模的可约性及相应的奇异向量.