The Six-Vertex Model at Roots of Unity and some Highest Weight Representations of the sl 2 Loop Algebra

We discuss irreducible highest weight representations of the sl₂ loop algebra and reducible indecomposable ones in association with the sl₂ loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation to be irreducible. We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality. Communicated by Vincent Rivasseau Dedicated to Daniel Arnaudon Submitted: March 3, 2006; Accepted: March 13, 2006     

Kac Determinant Formula for the q-deformed Virasoro Algebra of Hom-typ e

We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.     

Hom-型的q-形变Virasoro代数的Kac行列式公式（英文）

We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.     

Quasifinite modules of a Lie algebra related to Block type

Let B be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible B-module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight B-modules are classified, and the intermediate series B-modules are classified and constructed.     

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.     

The Polynomial Behavior of Weight Multiplicities for the Affine Kac–Moody Algebras A(1)r

We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A ⁽¹⁾ _r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.     

A character formula for a family of simple modular representations of $ GL_n $

Let K be an algebraically closed field of finite characteristic p, and let be an integer. In the paper, we give a character formula for all simple rational representations of with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight λ is special if there are integers such that and . Indeed, we compute the character of any simple module whose highest weight λ can be written as with all are special. By stabilization, we get a character formula for a family of irreducible rational -modules. Received: June 30, 1997.     

Classification of Simple Weight Modules for the Neveu–Schwarz Algebra with a Finite-Dimensional Weight Space

We show that the support of a simple weight module over the Neveu–Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu–Schwarz algebra, having a nontrivial finite-dimensional weight space, is a Harish–Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.     

Perfect Bases for Integrable Modules over Generalized Kac-Moody Algebras

We introduce the notion of perfect bases for integrable highest weight modules over generalized Kac-Moody algebras and show that the colored oriented graphs arising from perfect bases are isomorphic to the highest weight crystals B(λ) over quantum generalized Kac-Moody algebras.     

Classification of simple weight Virasoro modules with a finite-dimensional weight space

We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a simple highest or lowest weight module or a simple module from the intermediate series). This implies positive answers to two conjectures about simple pointed and simple mixed modules over the Virasoro algebra.     

Young wall realization of crystal bases for classical Lie algebras

In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

     

A character formula for infinite dimensional representations of the type i superalgebras sl(m/n) and c(2)

Infinite dimensional representations of Type I Lie superalgebra sl(m/n) for which the highest weight is singly or multiply and diagonally atypical are shown to have a character formula which is a modified version of the Bernstein-Leites character formula. This formula is also shown to be valid for all infinite dimensional representations of the Lie superalgebra C(2) classified according to the atypicality type of the highest weight A     

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.

     

Tensor products of modules with restricted highest weight

The summand containing the highest weight space of a tensor product of G-modules with restricted highest weights is studied.It is shown that for certain tensor products these summands are partial tilting modules for the algebraic group.     

Weight sets of irreducible integrable representations of kac-moody algebras of indefinite type

In this paper, the weight sets of some irreducible and integrable representations, which are not highest or lowest weight representations, of rank two Kac-Moody algebras of indefinite type are completely determined.     

Pointed representations of Virasoro algebra

In this paper we study the pointed representations of the Virasoro algebra. We show that unitary irreducible pointed representations of the Virasoro algebra are Harish-Chandra representations, thus they either are of highest or lowest weights or have all weight spaces of dimension 1. Further, we prove that unitary irreducible weight representations of Virasoro superalgebras are either of highest weights or of lowest weights, hence they are also Harish-Chandra representations. This work was supported by CNSF     

A Note on Weyl Modules for ∞ and ∞

In this article we extend Jantzen's formula for the determinant of the contravariant form on highest weight   _n -modules to a formula for highest weight   _∞-modules. This, in turn, has applications to the representation theory of (degenerate and nondegenerate) cyclotomic Hecke algebras.