Classification Of Simple Weight Modules Research Articles

Simple weight modules for Yangian Y (𝔰𝔩2)

Let [Formula: see text] be a finite-dimensional simple Lie algebra over [Formula: see text]. A [Formula: see text]-module is said to be weight if it is a weight [Formula: see text]-module. We give a complete classification of simple weight modules for [Formula: see text] which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical [Formula: see text]-representation theory, we show that there exists a class of [Formula: see text] irreducible modules which have uniformly two-dimensional weight spaces.

THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS

The rank n symplectic oscillator Lie algebra 𝔤n is the semidirect product of the symplectic Lie algebra 𝔰𝔭2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤n \( \left[\dot{z}\right] \) of the BGG category 𝒪𝔤n for 𝔤n and the BGG category 𝒪𝔰𝔭2n for 𝔰𝔭2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤n-modules (not necessarily weight modules) that have a simple Hn-submodule.

Classification of Simple Weight Modules over the Schrödinger Algebra

AbstractA classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer (H) (and some of its prime factor algebras) of the Cartan element H in the universal enveloping algebra of the Schrödinger (Lie) algebra. The simple (H)-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra (over the centre). It is proved that some (prime) factor algebras of and (H) are tensor homological/Krull minimal.

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

There are two extensions of Virasoro algebra with particular importance in superstring theory: the Ramond algebra and the Neveu-Schwarz algebra, which are ℤ2-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.

Simple weight modules over weak generalized Weyl algebras

In this paper we address the problem of classification of simple weight modules over weak generalized Weyl algebras of rank one. The principal difference between weak generalized Weyl algebras and generalized Weyl algebras is that weak generalized Weyl algebras are defined using an endomorphism rather than an automorphism of a commutative ring R. We reduce classification of simple weight modules over weak generalized Weyl algebras to description of the dynamics of the action of the above mentioned endomorphism on the set of maximal ideals. We also describe applications of our results to the study of generalized Heisenberg algebras.

Classification of Simple Weight Modules Over the 1-Spatial Ageing Algebra

In this paper we use Block’s classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra $\mathfrak {age(1)}$ . Most of these modules have infinite dimensional weight spaces and so far the algebra $\mathfrak {age(1)}$ is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrödinger algebra $\mathcal {S}$ that have a simple $\mathfrak {age(1)}$ -submodule thus constructing many new simple weight $\mathcal {S}$ -modules.

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.

Supports of weight modules over Witt algebras

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).