Highest weight representations of a Lie algebra of Block type

For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of the Block type is defined with the basis {Lα,i, c|α∈G, -1≤i∈Z} and the relations [Lα,i,Lβ,j] = ((i 1)β- (j 1)α)Lα β,i j αδα,-βδi j,-2c,[c, Lα,i] = 0. Given a total order (?) on G compatible with its group structure, and anyα∈B(G)0*, a Verma B(G)-module M(A, (?)) is defined, and the irreducibility of M(A,(?)) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z )-module is quasifinite if and only if it is a proper quotient of a Verma module.     

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on ＂polynomials＂ for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.     

Non-weight representations of Cartan type S Lie algebras

For the two Cartan type S subalgebras of the Witt algebra 𝒲_n, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra 𝔥_n. We also give all submodules of these modules.     

Twisted Hamiltonian Lie algebras and their multiplicity-free representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.     

Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Let B be the Block type Lie algebra over ? with basis {L _{α, i} , C ₁, C ₂ | (α, i) ∈ ? × ? \ {(0, ? 2)}} and Lie bracket [L _{α, i} , L _{β, j} ] = (β(i + 1) ? α(j + 1))L _{α+β, i+j}  + αδ_{α+β, 0}δ _{i+j, ?2} C ₁ + (i + 1)δ_{α+β, 0}δ _{i+j, ?2} C ₂, where C ₁, C ₂ are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.     

Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.      

Verma type modules of level zero for affine Lie algebras

We study the structure of Verma type modules of level zero induced from non-standard Borel subalgebras of an affine Kac-Moody algebra. For such modules in ``general position' we describe the unique irreducible quotients, construct a BGG type resolution and prove the BGG duality in certain categories. All results are extended to generalized Verma type modules of zero level.

     

On classification of quasifinite representations of two classes of Lie superalgebras of block type

Motivated by a well-known theorem of Mathieu’s on Harish–Chandra modules over the Virasoro algebra and its super version, we show that an irreducible quasifinite module over two classes of Lie superalgebras 𝒮(q) of Block type is either a highest or lowest weight module or else a module of the intermediate series if q≠?1. For such a module over 𝒮(?1), we give a rough classification.     

Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

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.     

Lie Bialgebras of a Family of Lie Algebras of Block Type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.     

Compatible actions of Lie algebras

Abstract

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XMod_L(Lie_R).     

一类Block型李代数的分类和中心扩张

本文我们决定了一类Block型李代数$\BB$在同构意义下的分类和中心扩张, 其中$q$是一个非零复数. 我们的结果推广了以前的一些结果.