For any nullity $2$ extended affine Lie algebra $mathcal{E}$ of maximal type and $ellinmathbb{C}$, we prove that there exist a vertex algebra $V_{mathcal{E}}(ell)$ and an automorphism group $G$ of $V_{mathcal{E}}(ell)$ equipped with a linear characte
r $chi$, such that the category of restricted $mathcal{E}$-modules of level $ell$ is canonically isomorphic to the category of $(G,chi)$-equivariant $phi$-coordinated quasi $V_{mathcal{E}}(ell)$-modules. Moreover, when $ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{mathcal{E}}(ell)$ of $V_{mathcal{E}}(ell)$ modulo by a $G$-stable ideal, and we prove that the integrable restricted $mathcal{E}$-modules of level $ell$ are exactly the $(G,chi)$-equivariant $phi$-coordinated quasi $L_{mathcal{E}}(ell)$-modules.
To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for vertex algebras, where $G$ is a group with $chi_phi$ a linear character of $G$ and $phi$ is an
associate of the one-dimensional additive formal group. The theory of $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for nonlocal vertex algebra is established in cite{JKLT}. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules. Furthermore, for any conformal algebra $mathcal{C}$, we construct a class of Lie algebras $widehat{mathcal{C}}_phi[G]$ and prove that restricted $widehat{mathcal{C}}_phi[G]$-modules are exactly $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for the universal enveloping vertex algebra of $mathcal{C}$. As an application, we determine the $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for affine and Virasoro vertex algebras.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vert
ex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $mathbb N$-graded irreducible modules by using a notion of Verma module. On
the other hand, we introduce a new algebra, a twin of the original algebra. Using this new algebra we construct a quantum vertex algebra and we associate $mathbb N$-graded modules for Jing-Nies Clifford-like algebra with $phi$-coordinated modules for the quantum vertex algebra. We also show that the adjoint module for the quantum vertex algebra is irreducible.
This is a paper in a series systematically to study toroidal vertex algebras. Previously, a theory of toroidal vertex algebras and modules was developed and toroidal vertex algebras were explicitly associated to toroidal Lie algebras. In this paper,
we study twisted modules for toroidal vertex algebras. More specifically, we introduce a notion of twisted module for a general toroidal vertex algebra with a finite order automorphism and we give a general construction of toroidal vertex algebras and twisted modules. We then use this construction to establish a natural association of toroidal vertex algebras and twisted modules to twisted toroidal Lie algebras. This together with some other known results implies that almost all extended affine Lie algebras can be associated to toroidal vertex algebras.
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 $S_c=U(L_r(mathbb{C}c))$. We furthermore study a distinguished subalgebra of $V(T,0)$, denoted by $V(S_c,0)$. We show that (graded) simple quotient toroidal vertex algebras of $V(S_c,0)$ are parametrized by a $mathbb{Z}^r$-graded ring homomorphism $psi:S_crightarrow L_r$ such that Im$psi$ is a $mathbb{Z}^r$-graded simple $S_c$-module. Denote by $L(psi,0}$ the simple $(r+1)$-toroidal vertex algebra of $V(S_c,0)$ associated to $psi$. We determine for which $psi$, $L(psi,0)$ is an integrable $L_{r}(hat{frak{g}})$-module and we then classify irreducible $L(psi,0)$-modules for such a $psi$. For our need, we also obtain various general results.
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $ell$, the category
of restricted $D$-modules of level $ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $ell$ is canonically isomorphic to the category of $mathbb{Z}$-equivariant $phi$-coordinated quasi modules for the same vertex algebra. In the process, we introduce and employ a certain infinite dimensional Lie algebra which is defined in terms of generators and relations and then identified explicitly with a subalgebra of $mathfrak{gl}_{infty}$.
Let $Gamma$ be a generic subgroup of the multiplicative group $mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Gamma$, called twisted $Gamma$-Lie algebras, which is a natural generalization of the twisted aff
ine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $mathbb Z^N$ and an arbitrary finite order isometry of $mathbb Z^N$ preserving $Q$, we construct a family of twisted $Gamma$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Gamma$-Lie algebras. As application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type $A$, trigonometric Lie algebras of series $A$ and $B$, unitary Lie algebras, and $BC$-graded Lie algebras.
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $tau$. More specifically, we define and study two categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$ of $tau$-modules using g
enerating functions, where $mathcal{E}_{tau}$ is proved to contain the evaluation modules while $mathcal{C}_{tau}$ contains certain restricted $tau$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$.
In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can
be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra $g$ we construct a vertex Leibniz algebra $V_{g}$ and show that $V_{g}$ can be embedded into a vertex algebra if and only if $g$ is a Lie algebra.
