Infinite-dimensional Weight Spaces Research Articles

Relaxed and logarithmic modules of sl3^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{{{\\mathfrak {s}}}{{\\mathfrak {l}}}_3}$$\\end{document}

In Adamović (Commun Math Phys 366:1025–1067, 2019), the affine vertex algebra Lk(sl2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k({{\\mathfrak {s}}}{{\\mathfrak {l}}}_2)$$\\end{document} is realized as a subalgebra of the vertex algebra Virc⊗Π(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Vir_c \\otimes \\Pi (0)$$\\end{document}, where Virc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Vir_c $$\\end{document} is a simple Virasoro vertex algebra and Π(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (0)$$\\end{document} is a half-lattice vertex algebra. Moreover, all Lk(sl2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k({{\\mathfrak {s}}}{{\\mathfrak {l}}}_2)$$\\end{document}-modules (including, modules in the category KLk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$KL_k$$\\end{document}, relaxed highest weight modules and logarithmic modules) are realized as Virc⊗Π(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Vir_c \\otimes \\Pi (0)$$\\end{document}-modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case g=sl3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {g}}= {{\\mathfrak {s}}}{{\\mathfrak {l}}}_3$$\\end{document} and present realization of the VOA Lk(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k({\\mathfrak {g}})$$\\end{document} for k∉Z≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \ otin {\\mathbb {Z}}_{\\ge 0}$$\\end{document} as a vertex subalgebra of Wk⊗S⊗Π(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {W}}_ k \\otimes {\\mathcal {S}} \\otimes \\Pi (0)$$\\end{document}, where Wk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {W}}_ k $$\\end{document} is a simple Bershadsky–Polyakov vertex algebra and S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {S}}$$\\end{document} is the βγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\gamma $$\\end{document} vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand–Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {g}}$$\\end{document}-modules which are not Gelfand–Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of Wk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {W}}_ k $$\\end{document} from Adamović et al. (Lett Math Phys 111(2), Paper No. 38, arXiv:2007.00396 [math.QA], 2021) and obtain a realization of logarithmic modules for Wk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {W}}_ k $$\\end{document} of nilpotent rank two at most admissible levels. Beyond admissible levels, we get realization of logarithmic modules up to a existence of certain Wk(sl3,fpr)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {W}}_k({{\\mathfrak {s}}}{{\\mathfrak {l}}}_3, f_{pr})$$\\end{document}-modules. Using logarithmic modules for the βγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\gamma $$\\end{document} VOA, we are able to construct logarithmic Lk(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k({\\mathfrak {g}})$$\\end{document}-modules of rank three.

Tensor product weight modules for the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible weight modules with infinite dimensional weight spaces over the mirror Heisenberg-Virasoro algebra D . More precisely, the necessary and sufficient conditions for the tensor products of irreducible highest weight modules and irreducible modules of intermediate series over D to be irreducible are determined by using “shifting technique”. This leads to a family of new irreducible weight modules over D . Then we obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and modules of intermediate series are isomorphic respectively. Also we discuss submodules of the tensor product module when it is not irreducible.

Representations of the q-Klein-bottle Lie algebra

We give the necessary and sufficient conditions for all weight spaces of the highest weight irreducible module V(ϕ) of the q-Klein-bottle Lie algebra B to be finite dimensional. We prove that the irreducible highest weight module V(ϕ) equals to the Verma B-module V¯(ϕ) if and only if V(ϕ) has at least one infinite-dimensional weight space. We also give the maximal proper submodule of the Verma module V¯(ϕ) and the e-character of the irreducible highest weight B-module V(ϕ) when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra B-modules with nontrivial central charge.

On representations of the centrally extended Heisenberg double of SL2

For the centrally extended Heisenberg double of SL2, its center is determined, the central factor algebras are described, and classifications of simple Harish-Chandra modules, simple Whittaker modules, and simple quasi-Whittaker modules are obtained. Two classes of simple weight modules with infinite-dimensional weight spaces are given. We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional sl2-modules with finite multiplicities.

Classification of simple Gelfand–Tsetlin modules of 𝔰𝔩(3)

We provide a classification and an explicit realization of all simple Gelfand–Tsetlin modules of the complex Lie algebra [Formula: see text]. The realization of these modules, including those with infinite-dimensional weight spaces, is given via regular and derivative Gelfand–Tsetlin tableaux. Also, we show that all simple Gelfand–Tsetlin [Formula: see text]-modules can be obtained as subquotients of localized Gelfand–Tsetlin [Formula: see text]-injective modules.

Irreducible modules over the divergence zero algebras and their [formula omitted]-analogues

In this paper, we study a class of Zd-graded modules, which are constructed using Shen–Larsson’s functor from sld-modules V, for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.

New families of irreducible weight modules over [formula omitted

Let n>1 be an integer, α∈Cn, b∈C, and V a gln-module. We define a class of weight modules Fbα(V) over sln+1 using the restriction of modules of tensor fields over the Lie algebra of vector fields on n-dimensional torus. In this paper we consider the case n=2 and prove the irreducibility of such 5-parameter sl3-modules Fbα(V) generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand–Tsetlin modules. Hence, this construction yields new families of irreducible sl3-modules.

Irreducible weight modules for the Virasoro-like algebra and its q-analog

ABSTRACTIn this paper, using Larsson’s functor with irreducible 𝔰𝔩2-modules V, we construct a class of ℤ2-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.

Generalized polynomial modules over the Virasoro algebra

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, \Omega(\lambda,\beta))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega(\lambda,\beta)$. We give necessary and sufficient conditions for the Virasoro module $F(M, \Omega(\lambda,\beta))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, \Omega(\lambda,\beta))$.

Irreducible A 1 (1) -modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules Fα(V) with zero central charge over the affine Lie algebra A1(1). These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules Fα(V) to be irreducible. In this way, we obtain a lot of irreducible weight A1(1)-modules with infinite-dimensional weight spaces.

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.

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.

A class of simple weight Virasoro modules

For a simple module M over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras ar) and any α∈C, a class of weight modules N(M,α) over the Virasoro algebra is constructed. These weight modules have infinite dimensional weight spaces if dim⁡M>1. The necessary and sufficient conditions for N(M,α) to be simple are obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules Γ(α1,α2,λ1,λ2) are defined on the polynomial algebra C[x]⊗C[t,t−1] for any α1,α2,λ1,λ2∈C with λ1 or λ2 nonzero. By twisting the weight modules N(M,α) we also obtain nonweight simple Virasoro modules N(M,β) for any nonconstant β∈C[t,t−1].

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.

Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and W(2, 2) algebra

We show that subsingular vectors may exist in Verma modules over W(2, 2), and present the subquotient structure of these modules. We prove conditions for irreducibility of the tensor product of intermediate series module with a highest weight module. Relation to intertwining operators over vertex operator algebra associated with W(2, 2) is discussed. Also, we study the tensor product of intermediate series and a highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.

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.

Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the Schrodinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.

Classification of irreducible weight modules over W-algebra W(2,2)

In this paper, we show that the support of an irreducible weight module over the W-algebra W(2,2), which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the W-algebra W(2,2), having a nontrivial finite dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or a lowest weight module or an irreducible module of the intermediate series).

Class of representations of skew derivation Lie algebra over quantum torus

Let L be the skew derivation Lie algebra of the quantum torus ℂq. In this paper, we give a class of irreducible representations for L with infinite dimensional weight spaces.

Classification of Irreducible Weight Modules with a Finite–dimensional Weight Space over Twisted Heisenberg–Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg– Virasoro algebra, which has an infinite–dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg–Virasoro algebra, having a nontrivial finite–dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series).