The Gelfand-Kirillov dimension of a unitary highest weight module

During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.

A Class of Simple Modules for Electrical Lie Algebra of Type D_(5)

The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.

Weight modules over some Block algebras

In this paper, the Harish-Chandra modules and Verma modules over Block algebra $ \mathfrak{L} $ [G] are investigated. More precisely, the irreducibility of the Verma modules over $ \mathfrak{L} $ [G] is completely determined, and the Harish-Chandra modules over $ \mathfrak{L} $ [?] are classified.

Tensor product weight modules of Schrodinger-Virasoro algebras

It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schr?dinger- Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schr?dinger-Virasoro algebra is obtained.

Structure and Representations for Electrical Lie Algebra of Type D_(5)

In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_(5).

Imaginary Modules over the Affine Nappi-Witten Algebra

In this paper,we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra.We show that simple singular imaginary Whittaker modules at level(κ、c)(κ∈C~*)are simple imaginary highest weight modules.The necessary and sufficient conditions for these imaginary modules to be simple are given.All simple imaginary modules are classified.

Grbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G_2

First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirshov basis of Uq（G2）. Then, by specializing a suitable version of Uq （G2） at q = 1, they get a GrSbner-Shirshov basis of the universal enveloping algebra U（G2） of the simple Lie algebra of type G2 and the finite-dimensional irreducible U（G2）-module V（λ）.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.

Determinant formula and a realization for the Lie algebra W(2,2)

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).