Toroidal Lie Algebras Research Articles

Bigraded modified Toda hierarchy and its extensions

Modified Toda hierarchy is just the two-component first modified KP hierarchy, which is related to 2D Toda hierarchy through Miura transformation and also has been widely used in discussing the B-Toda and C-Toda hierarchies. In this paper, we firstly construct (N,M)-bigraded modified Toda hierarchy (BMTH) as a reduction of modified Toda hierarchy, and give corresponding Lie algebra interpretation. After that, we propose two kinds of extensions of (N,M)-BMTH. One is extended by using logarithmic flows, while the other is (2+1)D extension, which is corresponding to the toroidal Lie algebra slntor. At last, the relation of these two kinds of extensions also is discussed.

Representations of toroidal and full toroidal Lie algebras over polynomial algebras

Toroidal Lie algebras are n variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of n-variable generalization of Affine-Virasoro algebras. Let h̃ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on U(h̃) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U(h̃)-modules of rank 1. These modules exist only for type Al(l ≥ 1), Cl(l ≥ 2) toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.

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.

Vertex algebras and TKK algebras

In this paper, we associate the TKK algebra Gˆ(J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ, the category of restricted Gˆ(J)-modules of level ℓ is canonically isomorphic to the category of σ-twisted VCgˆ(ℓ,0)-modules, where VCgˆ(ℓ,0) is a vertex algebra arising from the toroidal Lie algebra of type C2 and σ is an isomorphism of VCgˆ(ℓ,0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ, the integrable restricted Gˆ(J)-modules of level ℓ are exactly the σ-twisted modules for the quotient vertex algebra LCgˆ(ℓ,0) of VCgˆ(ℓ,0). Finally, we classify the irreducible 12N-graded σ-twisted LCgˆ(ℓ,0)-modules.

Howe duality in the toroidal setting

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup.

Weyl Modules for Toroidal Lie Algebras

In this paper, we study Weyl modules for a toroidal Lie algebra \(\mathcal {T}\) with arbitrary n variables. Using the work of Rao (Pac. J. Math. 171(2), 511–528 1995), we prove that the level one global Weyl modules of \(\mathcal {T}\) are isomorphic to suitable submodules of a Fock space representation of \(\mathcal {T}\) up to a twist. As an application, we compute the graded character of the level one local Weyl module of \(\mathcal {T}\), thereby generalising the work of Kodera (Lett. Math. Phys. 110(11) 3053–3080 2020).

Irreducible modules of toroidal Lie algebras arising from ϕϵ-coordinated modules of vertex algebras

In this paper, for every ϵ∈Z, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the ϕϵ-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When ϵ=1, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

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.

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

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.

Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

Borel–de Siebenthal Theory for Affine Reflection Systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

Integrable lattice models and holography

We study four-dimensional Chern-Simons theory on D × ℂ (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an “analytically-continued” toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.

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.

Vertex algebras and extended affine Lie algebras coordinated by rational quantum tori

Let slNˆ(Cq) be the core of extended affine Lie algebra of type AN−1 coordinated by the rational quantum 2-torus Cq. In this paper, we first prove that for any complex number ℓ, the category of restricted slNˆ(Cq)-modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra VCgˆ(ℓ,0) arising from a conformal Lie algebra Cg, where Cgˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ, the integrable restricted slNˆ(Cq)-modules of level ℓ are exactly the twisted modules for the quotient vertex algebra LCgˆ(ℓ,0) of VCgˆ(ℓ,0). Finally, we classify irreducible graded twisted LCgˆ(ℓ,0)-modules.

Drinfeld-type presentations of loop algebras

Let g \mathfrak {g} be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ \mu be a diagram automorphism of g \mathfrak {g} , and let L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) be the loop algebra of g \mathfrak {g} associated to μ \mu . In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] of L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) . The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] as special examples. As an application, when g \mathfrak {g} is of simply-laced-type, we prove that the classical limit of the μ \mu -twisted quantum affinization of the quantum Kac-Moody algebra associated to g \mathfrak {g} introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] .

Level one Weyl modules for toroidal Lie algebras

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 by-product, 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.

Fermionic realization of twisted toroidal Lie algebras

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type [Formula: see text] and [Formula: see text] based on the newly found Moody–Rao–Yokonuma-like presentation.

Classification of bounded modules for the twisted full toroidal Lie algebras

In this article, we study modules for 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 is explicitly constructed by Michael Lau and Yuly Billig using Vertex Operator Algebras. We show that they exhaust all the irreducible bounded modules. We prove that the top spaces of both these classes of modules are the same and hence the corresponding irreducible modules are also the same.

Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation

Let $\fg$ be an affine Kac-Moody algebra, and $\mu$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[\mu]$ of the twisted loop algebra of $\fg$ related to $\mu$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[\mu]$ when $\mu$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.