We study irreducible modules for Toroidal Lie-algebras with finite dimensional weight spaces. First note that Toroidal Lie-algebras have infinite dimensional center. In genaral the infinite dimensional center does not act as scalars on an irreducible module but acts as invertible central operators. We prove in this paper the study of irreducible modules for Toroidal Lie-algebras with finite dimensional weight spaces is reduced to the study of irreducible modules where the center acts as scalars.
In this paper we classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. In both the cases we classify modules, when a part of center acts non-trivially and trivially on modules.
The irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras on which the center acts non-trivially were classified by S.Eswara Rao. In this paper we give a compact proof of the results that lead to the classification theorem. Further we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.
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.
Toroidal Lie algebras are universal central extentions of the finite dimensional simple Lie algbera tensored with Laurent Polynomials in several commuteing variables. In this paper we classify irreducible integrable modules for Toroidal Lie algebras with finite dimensional wieght spaces. They are tensor product of evaluation modules for the finite dimensional Lie algebra when the center acts by zero. In the case when the center acts nontrivially they are tensor product of evalution representations of the underlying affine Lie algebra twisted by an automorphism of the Toroidal Lie algebras. The…
In this paper, we study modules for the Twisted Full Toroidal Lie Algebras (TFTLA). We define a category of bounded modules for TFTLA and classify all the irreducible modules in that category. A class of irreducible bounded modules are explicitly constructed by Michael Lau and Yuly Billig using Vertex operator algebras. We show that they exhaust all the irreducible bounded modules. Our plan is to describe the top spaces of the bounded irreducible modules for TFTLA and prove they in fact coincide with the top spaces of the modules constructed by Lau and Billig. When the top spaces become isomor…
We show that the representation theory for the toroidal extended affine Lie algebras is controlled by a vertex operator algebra which is a tensor product of four VOAs: a sub-VOA of the hyperbolic lattice VOA, two affine VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N=12, the sub-VOA of the hyperbolic lattice VOA becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.
Let $\mathbb{C}_q$ be a non-commutative Laurent polynomial ring associated with a $(n+1)\times (n+1)$ rational quantum matrix $q$. Let $\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ be the universal central extension of Lie subalgebra $\mathfrak{sl}_d(\mathbb{C}_q)$ of $\mathfrak{gl}_d(\mathbb{C}_q)$. Now let us take the Lie algebra $τ=\mathfrak{gl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$. Let $Der(\mathbb{C}_q)$ be the Lie algebra of all derivations of $\mathbb{C}_q$. Now we consider the Lie algebra $\tildeτ=τ\rtimes Der(\mathbb{C}_q)$, called as full toroidal Lie algebra co-ordin…
We introduce a category B of bounded modules for the toroidal Lie algebras and study irreducible modules in B. We show that one of the irreducible modules in this category, L(T_0), admits a structure of a vertex operator algebra. We prove that L(T_0) factors into a tensor product of a sub-VOA of a hyperbolic lattice VOA and a simple VOA associated with a twisted Virasoro-affine Lie algebra. Every irreducible module in category B is a VOA module for a slightly larger VOA V(T_0). Knowing the structure of V(T_0), we are able to give explicit realizations for all irreducible modules in category B …
Toroidal Lie algebras are generalizations of affine Lie algebras. In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras. In this paper we give a presentation for the twisted toroidal Lie algebras of type $A$ and $D$ constructed by Fu and Jiang.
Using $n$ finite order automorphisms on a simple complex Lie algebra we construct twisted $n$-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case $n=2$ we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Saito.
This is an expository article on representation theory of toroidal Lie algebras. We summerize all the results on representation theory of toroidal Lie algebras obtained in the last fifteen years. Apart from that a natural genaralization of Virasoro Lie algebra is introduced and the related results are stated. A genaralization of the semi direct product of affine Lie algebra and Virasoro Lie algebra is introduced which we call the full toroidal Lie algebra. The connection to the theory of Extended Affine Lie Algebras is explained.
In this paper we construct a large class of modules for toroidal Lie superalgebras. Toroidal Lie superalgebras are universal central extension of G tensor A where G is a basic classical Lie superalgebra and A is a Laurent polynomial ring in sevaral variables. The case where G is simple finite dimensional Lie algebra is included.
The paper is to classify irreducible integrable modules for the twisted full toroidal Lie algebra with some technical conditions. The twisted full toroidal Lie algebra are extensions of multiloop algebra twisted by sevaral finite order automorphisms. The result genaralizes a result by Fu Jiayuan and Cuipo Jiang, where they consider only one automorphism.
In this paper we study the subcategory of finite-length objects of the category of positive level integrable representations of a toroidal Lie algebra. The main goal is to characterize the blocks of the category. In the cases when the underlying finite type Lie algebra associated with the toroidal Lie algebra is simply-laced, we are able to give a parametrization for the blocks.
In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where $\widetilde{\mathfrak g}_c$ is the core of $\widetilde{\mathfrak g}$.
Using free fields, we construct a bosonic realization of toroidal Lie algebras of type $A_{2n-1}, A_{2n}, D_{n+1}, D_4$ which are twisted by a Dynkin diagram automorphism. This realization is based upon the recently found Moody-Rao-Yokonuma-like presentation. In particular, our construction contains the first bosonic realization of the twisted affine algebra $D_4^{(3)}$.
In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type $A_{2n-1}, D_{n+1}, A_{2n}$ and $D_4$ based on the newly found Moody-Rao-Yokonuma-like presentation.
The Toroidal Lie algebras are n variable genaralizations of Affine Kac-Moody Lie algebras. As in the affine Lie algebras there exists finite order auto= morphisms corresponding to Dynkin diagram automorphisms. The fixed point sub= algebras are called twisted toroidal Lie algebras. In this paper we construt faithfull representations for the twisted toroidal Lie algebras (this includes the non-twisted case also) useing methods developed by Lepowsky-Wilson [LW]. This construction recovers the result by Eswara Rao-Moody [EM] in the homogeneous picture and by Yuly Billig [B1] in the principal pictu…
In this paper we study Weyl modules for a toroidal Lie algebra $\CT$ with arbitrary $n$ variables. Using the work of Rao \cite{1995}, we prove that the level one global Weyl modules of $\CT$ are isomorphic to suitable submodules of a Fock space representation of $\CT$ upto a twist. As an application, we compute the graded character of the level one local Weyl module of $\CT$, thereby generalising the work of Kodera \cite{ko}.
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $τ$. More specifically, we define and study two categories $\mathcal{E}_τ$ and $\mathcal{C}_τ$ of $τ$-modules using generating functions, where $\mathcal{E}_τ$ is proved to contain the evaluation modules while $\mathcal{C}_τ$ contains certain restricted $τ$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $\mathcal{E}_τ$ and $\mathcal{C}_τ$.
Let $\fg$ be any untwisted affine Kac-Moody algebra, $μ$ any fixed complex number, and $\wt\fg(μ)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bmλ=(λ_1, \cdots, λ_k)$ of weights of $\fg$, and $k$-tuple $\bm{a}=(a_1,\cdots, a_k)$ of distinct non-zero complex numbers, we construct a class of modules $\wt V(\bmλ,\bm{a})$ for the extended affine Lie algebra $\wt\fg(μ)$. We prove that the $\wt\fg(μ)$-module $\wt V(\bmλ,\bm{a})$ is completely reducible. We also prove that the $\wt\fg(μ)$-module $\wt V(\bmλ,\bm{a})$ is integrable when all weights $λ_i$ i…
In this paper, we continue the study on toroidal vertex algebras initiated in \cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$, where $\hat{\frak{g}}$ is an untwisted affine Lie algebra and $L_r=$\mathbb{C}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$. We first construct an $(r+1)$-toroidal vertex algebra $V(T,0)$ and show that the category of restricted $L_{r}(\hat{\frak{g}})$-modules is canonically isomorphic to that of $V(T,0)$-modules.Let $c$ denote the standard central element of $\hat{\frak{g}}$ and set $S…
We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.
We classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with fnite diemnsional weight spaces where the fnite dimensional center acts trivially. We have proved that the entire central extension part acts trivially on the modules and our modules turn out to be highest weight modules.
We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is given by an action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the toroidal Lie algebra. As a byproduct, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.
We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.
Generalizing Feingold-Frenkel's construction we use Weyl bosonic fields to construct toroidal Lie algebras of types $A_n, B_n$, $C_n$ and $D_n$ of level $-1, -2, -1/2$ and -2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.
We construct vertex operator representations for the full (N+1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal Lie algebra g. We also construct irreducible modules for an important subalgebra gdiv of the toroidal Lie algebra that corresponds to the divergence free vector fields. This subalgebra carries a non-degenerate invariant bilinear form. The VOA that co…
We consider general linear superalgebra (type A) and tensor with Laurent polynomial ring in several variables. We then consider the universal central extension of this Lie superalgebra which we call toroidal superalgebra. We give a faithful representation of toroidal superalgebra using vertex operators and bosons.
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.
We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some approaches which are used in the study of other well--known non--semisimple categories in the representation theory of Lie algebras. This is done with a view to seeing if and how far these approaches can be made to work for $\cal I$. In the later sections we focus first on understanding the irreducible level zero modules and later on certain universal modules, the …
In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.
Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.
Let $\fg$ be an affine Kac-Moody algebra, and $μ$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[μ]$ of the twisted loop algebra of $\fg$ related to $μ$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[μ]$ when $μ$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.
The description of irreducible finite dimensional representations of finite dimensional solvable Lie superalgebras over complex numbers given by V.~Kac is refined. In reality these representations are not just induced from a polarization but twisted, as infinite dimensional representations of solvable Lie algebras. Various cases of irreducibility (general and of type Q) are classified.
We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $ϕ:\mathbb N\rightarrow \{\pm\}$ we associate a $ϕ$-highest weight module over the Heisen…
For a twisted affine Lie superalgebra with nonzero odd part, we study {tight irreducible weight modules} with bounded weight multiplicities and show that if the action of nonzero real vectors of each affine component of the zero part is neither completely injective nor completely locally nilpotent, then these modules are parabolically induced.
In the present paper we classify all irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type W. The classification is based on a bijective correspondence between the continuous representations of the n-Lie algebras W^n and continuous representations of the Lie algebra of Cartan type W_{n-1}, on which some two-sided ideal acts trivially.
Linear Lie Groups All of Whose Irreducible Finite-Dimensional Not Necessarily Unitary Representations Are of Bounded Dimension and Separate the Points of the Group
In the present paper we classify all irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type S. The classification is based on a bijective correspondence between the continuous representations of the n-Lie algebras S^n and continuous representations of the Lie algebra of Cartan type S, on which some two-sided ideal acts trivially.
In this paper, we introduce the notion of completely non-trivial module of a Lie conformal algebra. By this notion, we classify all finite irreducible modules of a class of $\mathbb{Z}^+$-graded Lie conformal algebras $\mathcal{L}=\bigoplus_{i=0}^{\infty} \mathbb{C}[\partial]L_i$ satisfying $ [{L_0}_λL_0]=(\partial+2λ)L_0,$ and $[{L_1}_λL_i]\neq 0$ for any $i\in \mathbb{Z}^+$. These Lie conformal algebras include Block type Lie conformal algebra $\mathcal{B}(p)$ and map Virasoro Lie conformal algebra $\mathcal{V}(\mathbb{C}[T])=Vir\otimes \mathbb{C}[T]$. As a result, we show that all non-trivi…
In this note, we address a question raised by Krötz on the classification of domains of holomorphy of irreducible admissible Banach representations for connected non-compact simple real Lie groups G. When G is not of Hermitian type, we give a complete description of the domains of holomorphy for irreducible admissible uniformly bounded representations on uniformly convex uniformly smooth Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine the domains of holomorphy only when the representations considered a…
In the present paper, we introduce a class of infinite Lie conformal superalgebras $\mathcal{S}(p)$, which are closely related to Lie conformal algebras of extended Block type defined in \cite{CHS}. Then all finite non-trivial irreducible conformal modules over $\mathcal{S}(p)$ for $p\in\C^*$ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras $\mathfrak{s}(n)$ for $n\geq1$ and $\mathfrak{sh}$ which is isomorphic to a subalgebra of Lie conformal algebra of $N=2$ superconformal algebr…
This paper investigates the irreducibility of certain representations for the Lie algebra of divergence zero vector fields on a torus. In "Irreducible Representations of the Lie-Algebra of the Diffeomorphisms of a d-Dimensional Torus," S. Eswara Rao constructs modules for the Lie algebra of polynomial vector fields on a d-dimensional torus, and determines the conditions for irreducibility. The current paper considers the restriction of these modules to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie …
The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their representations, quaternary algebras and their alloyability are discussed. Considered examples allow to conclude that new representations have some intriguing features (continuous moduli of finite-dimensional irreducible representations, sophisticated Clebsch-Gordan coefficient calculus, etc.).
