Whittaker Module Research Articles

Whittaker modules over the N = 2 Neveu–Schwarz algebra

In this paper, we study Whittaker modules over the [Formula: see text] Neveu–Schwarz algebra [Formula: see text]. We first classify all simple finite-dimensional modules over the positive part of [Formula: see text]. Motivated by this classification, we define Whittaker modules and investigate their simplicity. Finally, we also study the simplicity of the quotient modules of the universal Whittaker module if it is not simple.

Representations of the affine ageing algebra agê(1)

In this paper, we investigate the affine ageing algebra agê(1), which is a central extension of the loop algebra of the one-spatial ageing algebra age(1). Certain Verma-type modules including Verma modules and imaginary Verma modules of agê(1) are studied. Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of agê(1) which are also the modules of the affine vertex algebra arising from the one-spatial ageing algebra age(1). We present certain constructions of simple restricted agê(1)-modules and an explicit such example of simple restricted module via the Whittaker module of agê(1) is given.

Classical Whittaker modules for the classical affine Kac-Moody algebras

Inspired by Sugawara operators, we introduce a new basis for the universal non-degenerate Whittaker module of level κ over the classical affine Kac-Moody algebra of type BN(1), CN(1) or DN(1). This basis is different from the one derived from the PBW basis and would help us to understand the structure of Whittaker modules. As a result, we classify simple non-degenerate Whittaker modules for the classical affine Kac-Moody algebras whether at the critical or noncritical level.

Whittaker modules and quasi-Whittaker modules for the Schrödinger algebra in (2 + 1)-dimensional spacetime

In this paper, we classify simple (quasi)-Whittaker modules for the Schrödinger Lie algebra s2 in (2+1)-dimensional space-time. Moreover, we show that a simple sn-module is a Whittaker module if and only if it is a locally finite sn+-module, and it is a quasi-Whittaker module if and only if it is a locally finite hn-module. Also, we determine the center of U(s2) and U(s2/Cz).

Affine Lie algebra representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.

Whittaker modules for the planar Galilean conformal algebra and its central extension

Let be the planar Galilean conformal algebra and be its universal central extension. Then (resp. ) admits a triangular decomposition: (resp. ). In this paper, we study universal and generic Whittaker -modules (resp. -modules) of type where is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker module of type is simple if and only if is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some proper submodules of generic Whittaker modules.

Whittaker category for the Lie algebra of polynomial vector fields

For any positive integer n, let An=C[t1,…,tn], Wn=Der(An) and Δn=Span{∂∂t1,…,∂∂tn}. Then (Wn,Δn) is a Whittaker pair. A Wn-module M is called a Whittaker module if the action of Δn on M is locally finite. We show that each block ΩaW˜ of the category of (An,Wn)-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over Ln, where Ln is the Lie subalgebra of Wn consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker Wn-modules with finite dimensional Whittaker vector spaces using gln-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ΩaW.

Imaginary Whittaker Modules of the Twisted Affine Nappi-Witten Lie Algebra <img src="http://admin.scirp.org/Editer/attached/image/Edit_fc740d60-7009-4d6f-939a-ac2a113f1539.bmp" alt="" />

The Nappi-Witten Lie algebra was first introduced by C. Nappi and E. Witten in the study of Wess-Zumino-Novikov-Witten (WZNW) models. They showed that the WZNW model (NW model) based on a central extension of the two-dimensional Euclidean group describes the homogeneous four-dimensional space-time corresponding to a gravitational plane wave. The associated Lie algebra is neither abelian nor semisimple. Recently K. Christodoulopoulou studied the irreducible Whittaker modules for finite- and infinite-dimensional Heisenberg algebras and for the Lie algebra obtained by adjoining a degree derivation to an infinite-dimensional Heisenberg algebra, and used these modules to construct a new class of modules for non-twisted affine algebras, which are called imaginary Whittaker modules. In this paper, imaginary Whittaker modules of the twisted affine Nappi-Witten Lie algebra are constructed based on Whittaker modules of Heisenberg algebras. It is proved that the imaginary Whittaker module with the center acting as a non-zero scalar is irreducible.

Whittaker modules for the twisted affine Nappi-Witten Lie algebra [formula omitted

The Whittaker module Mψ and its quotient Whittaker module Lψ,ξ for the twisted affine Nappi-Witten Lie algebra Hˆ4[τ] are studied. For nonsingular type, it is proved that if ξ≠0, then Lψ,ξ is irreducible and any irreducible Whittaker Hˆ4[τ]-module of type ψ with k acting as a non-zero scalar ξ is isomorphic to Lψ,ξ. Furthermore, for ξ=0, all Whittaker vectors of Lψ,0 are completely determined. For singular type, the Whittaker vectors of Lψ,ξ with ξ≠0 are fully characterized.

Arakawa-Suzuki functors for Whittaker modules

In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type An to the category of finite-dimensional modules of the graded affine Hecke algebra of type Aℓ. Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category O as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of SLn(C) and representations of the symmetric group Sn.

Simple Whittaker modules over free bosonic orbifold vertex operator algebras

We construct weak (i.e. nongraded) modules over the vertex operator algebra M ( 1 ) + M(1)^+ , which is the fixed-point subalgebra of the higher rank free bosonic (Heisenberg) vertex operator algebra with respect to the − 1 -1 automorphism. These weak modules are constructed from Whittaker modules for the higher rank Heisenberg algebra. We prove that the modules are simple as weak modules over M ( 1 ) + M(1)^+ and calculate their Whittaker type when regarded as modules for the Virasoro Lie algebra. Lastly, we show that any Whittaker module for the Virasoro Lie algebra occurs in this way. These results are a higher rank generalization of some results by Tanabe [Proc. Amer. Math. Soc. 145 (2017), no. 10, pp. 4127–4140].

Whittaker modules for the affine Lie algebra [formula omitted

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra sl2ˆ of type A1(1) with noncritical level. These modules can become simple Whittaker modules over sl2˜=sl2ˆ+Cd with the same Whittaker function and central charge. We have to modulo a central character for sl2 to obtain simple degenerate Whittaker sl2ˆ-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker sl2ˆ-module by a submodule generated by a scalar action of central elements of the vertex algebra V−2(sl2) is simple as sl2ˆ-module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for sl2˜. Quite surprisingly, with the same Whittaker function some simple degenerate sl2˜ Whittaker modules at the critical level have semisimple action of d and others have free action of d. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for sl2ˆ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for sl2ˆ at noncritical level.

Whittaker modules and quasi-Whittaker modules for the Euclidean Lie algebra [formula omitted

In this paper, we classify simple Whittaker modules and simple quasi-Whittaker modules for the Euclidean Lie algebra E=e(3). We show that a simple e(3)-module is a Whittaker module if and only if it is a locally finite E+-module, and it is a quasi-Whittaker module if and only if it is a locally finite P-module. In particular, we get a class of simple weight e(3)-modules which have infinite-dimensional weight spaces.

Irreducible Virasoro modules from tensor products

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

Simple Schrödinger modules which are locally finite over the positive part

It is proved that a simple module which is locally finite over the positive part of the Cartan decomposition for a finite-dimensional Lie algebra with triangular decomposition and solvable positive part is a highest weight module, a Whittaker module or a singular Whittaker module. The classification of simple highest weight modules for the Schrödinger algebra follows from the classification of all simple lowest weight module for the Schrödinger algebra given by Dobrev, Doebner and Mrugalla. The simple Whittaker modules for the Schrödinger algebra are classified in this paper. The singular Whittaker vectors for the universal singular Whittaker module are given.

Whittaker modules over loop Virasoro algebra

In this paper, we study Whittaker modules over the loop Virasoro algebra relative to some total order. We give a description of all Whittaker vectors for the universal Whittaker modules. We also show that any universal Whittaker module admits a unique simple quotient modules except for a special case.

Whittaker Modules over Generalized Virasoro Algebras

In this article, we study Whittaker modules over generalized Virasoro algebras relative to some total order. We give the description of any Whittaker module and its Whittaker vectors. In particular, we determine all simple Whittaker modules, which provide the first known examples of simple nonweight modules over generalized Virasoro algebras. On the other hand, we also show that any Whittaker module over any generalized Virasoro algebra can be induced from a Whittaker module over some Virasoro subalgebra.

Tensor Products and Whittaker Vectors for Quantum Groups

We describe the action of the center of the quantum group Uq (𝔤) on the tensor product V ⊗ L(λ) of an infinite-dimensional representation V having an infinitesimal character χτ and an irreducible finite-dimensional Uq (𝔤) representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra Uq(𝔰l2).

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A 1 and finding interesting families of new noetherian rings, a class of algebras similar to U(sl 2) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.

Whittaker modules for [formula omitted

We define and study the class of Whittaker modules for the quantum enveloping algebra U q ( sl 2 ) of sl 2 . One of our main results describes an arbitrary Whittaker module as a quotient of U q ( sl 2 ) . From this description, we determine precise criteria for when a Whittaker module is simple as well as a decomposition of an arbitrary Whittaker module into indecomposable submodules. We also prove that the annihilator ann U q ( sl 2 ) ( V ) of a Whittaker module V is generated by its intersection with the center of U q ( sl 2 ) . This is the analogue of a classical result in the Lie algebra setting due to Kostant.