Finite Dimensional Weight Spaces Research Articles

Simple modules for twisted Hamiltonian extended affine Lie algebras

In this paper we consider the twisted Hamiltonian extended affine Lie algebra (THEALA). We classify the irreducible integrable modules for these Lie algebras with finite dimensional weight spaces when the finite dimensional center acts non-trivially. This Lie algebra has a triangular decomposition, which is different from the natural triangular decomposition of twisted full toroidal Lie algebra. Any irreducible integrable module of it is a highest weight module with respect to the given triangular decomposition. In this paper we describe the highest weight space in detail.

Representations of map extended Witt algebras

In this paper, we classify irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n≥2). So they are actually irreducible modules for extended Witt algebras.

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.

Dirac cohomology and orthogonality relations for weight modules

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincar\'{e} pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun.

Irreducible integrable modules for the full toroidal Lie algebras coordinated by rational quantum torus

Let Cq be the non-commutative Laurent polynomial ring associated with a (n+1)×(n+1) rational quantum matrix q. Let sld(Cq)⊕HC1(Cq) be the universal central extension of Lie subalgebra sld(Cq) of gld(Cq). We take the Lie algebra τ=gld(Cq)⊕HC1(Cq). Let Der(Cq) be the Lie algebra of all derivations of Cq and consider the Lie algebra τ˜=τ⋊Der(Cq), called the full toroidal Lie algebra coordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for τ˜ with nonzero central action on the modules.

Unitary Vertex Algebras and Wightman Conformal Field Theories

We prove an equivalence between the following notions: (i) unitary Möbius vertex algebras, and (ii) Wightman conformal field theories on the circle (with finite-dimensional conformal weight spaces) satisfying an additional condition that we call uniformly bounded order. Reading this equivalence in one direction, we obtain new analytic and operator-theoretic information about vertex operators. In the other direction we characterize OPEs of Wightman fields and show they satisfy the axioms of a vertex algebra. As an application we establish new results linking unitary vertex operator algebras with conformal nets.

Irreducible modules for the loop of derivations of rational quantum torus

Let [Formula: see text] be a rational quantum torus associated with the matrix [Formula: see text]. Let [Formula: see text] be the Lie algebra of derivations of [Formula: see text]. In this paper, we consider the Lie algebra [Formula: see text], where [Formula: see text] is a commutative associative unital algebra over [Formula: see text] and classify its irreducible modules with finite-dimensional weight spaces.

Representations of the Nappi–Witten vertex operator algebra

The Nappi-Witten model is a Wess-Zumino-Witten model in which the target space is the nonreductive Heisenberg group $H_4$. We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $\mathsf{H}_4$. In particular, we classify the irreducible $\mathsf{H}_4$-modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic conformal field theory.

Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus LT which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite-dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial LT-action and where the centre of the underlying Lie algebra acts trivially.

Irreducible modules over the mirror Heisenberg–Virasoro algebra

In this paper, we study irreducible modules over the mirror Heisenberg–Virasoro algebra [Formula: see text], which is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all Harish-Chandra modules over [Formula: see text], i.e. irreducible modules with finite-dimensional weight spaces. Every such module is either an irreducible highest or an irreducible lowest weight module, or an irreducible module of the intermediate series. Furthermore, we use a twisted version of Feigin–Fuchs construction of the Virasoro algebra to establish the simplicity criterion for Verma modules and obtain a classification of unitary irreducible highest weight modules over [Formula: see text]. Finally, we determine all irreducible restricted [Formula: see text]-modules of level zero.

Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras

The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.

Irreducible weight modules over the Schrödinger Lie algebra in (n + 1) dimensional space-time

In this paper, we study weight representations over the Schrödinger Lie algebra sn for any positive integer n. It turns out that the algebra sn can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight sn-modules with finite dimensional weight spaces for any n. All such modules can be clearly characterized by the tensor product of son-modules, sl2-modules and modules over the Weyl algebra.

Simple weight modules with finite-dimensional weight spaces over Witt superalgebras

Let Am,n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C, and the Witt superalgebra Wm,n be the Lie superalgebra of superderivations of Am,n. In this paper, we classify the simple weight Wm,n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of Wm,0. Every such module is either a simple quotient of a tensor module or a module of highest weight type.

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.

Integrable modules for graded Lie tori with finite-dimensional weight spaces

An important problem in the representation theory of affine and toroidal Lie algebras is to classify all possible irreducible integrable modules with finite-dimensional weight spaces. Recently the irreducible integrable modules having finite-dimensional weight spaces with non-trivial central action have been classified for a more general class of Lie algebras, namely the graded Lie tori. In this paper, we classify all the irreducible integrable modules with finite-dimensional weight spaces for this graded Lie tori where the central elements act trivially. Thus we ultimately obtain all the simple objects in the category of integrable modules with finite-dimensional weight spaces for the graded Lie tori.

Integrable modules for twisted toroidal extended affine Lie algebras

In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras (twisted toroidal EALA, in short) with finite dimensional weight spaces when the finite dimensional center acts non-trivially. Using an automorphism we reduce to the case where K0 acts non-trivially and Ki, 1≤i≤n act trivially. Twisted toroidal EALA has a natural triangular decomposition and we prove that any irreducible integrable module of it with finite dimensional weight spaces is a highest weight module with respect to the above triangular decomposition. The highest weight space is an irreducible module for the zeroth component of the twisted toroidal EALA. We then describe the highest weight space in detail.

Modules for loop Affine-Virasoro algebras

In this paper, we study the representations of loop Affine-Virasoro algebras. As they have canonical triangular decomposition, we define Verma modules and their irreducible quotients. We give necessary and sufficient condition for a irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either a highest weight module or a lowest weight module whenever the canonical central element acts non-trivially. At the end, we construct Affine central operators for each integer and they commute with the action of the Affine Lie algebra.

Harish-Chandra modules for divergence zero vector fields on a torus

Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized highest weight (GHW for short) module. Furthermore, we prove that the nonzero level Harish-Chandra modules of $\widetilde{L}$ are GHW modules. Finally, we classify all the GHW Harish-Chandra modules of $\widetilde{L}$.

Modules of highest weight type for the algebra of derivations over a rational quantum torus

For the Lie algebra of derivations over a rational quantum torus, we construct a class of irreducible weight modules with all finite dimensional weight spaces, which are called the modules of highest weight type, generating the notion for the case of commutative torus.

Irreducible Weight Modules with a Finite-Dimensional Weight Space over the Twisted N = 1 Schrödinger-Neveu-Schwarz Algebra

It is shown that there are no simple mixed modules over the twisted N = 1 Schrödinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight space is a Harish-Chandra module.