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In this paper, we associate the quantum toroidal algebra $\mathcal{E}_N$ of type $\mathfrak{gl}_N$ with quantum vertex algebra through equivariant $\phi$-coordinated quasi modules. More precisely, for every $\ell\in \mathbb{C}$, by deforming the universal affine vertex algebra of $\mathfrak{sl}_\infty$, we construct an $\hbar$-adic quantum $\Z$-vertex algebra $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$. Then we prove that the category of restricted $\mathcal{E}_N$-modules of level $\ell$ is canonically isomorphic to that of equivariant $\phi$-coordinated quasi $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$-modules.

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.

Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric properties of their Weyl groups; they possess the so-called ``presentation by conjugation". In this work, we characterize minimal extended affine root systems in terms of ``minimal reflectable bases" which resembles the concept of the ``base" for finite and affine root systems. As an application, we construct elliptic Lie algebras by means of Serre's type generators and relations., Comment: 31 pages

This paper is about establishing a natural connection of quantum affine algebras with quantum vertex algebras. Among the main results, we establish $\hbar$-adic versions of the smash product construction of quantum vertex algebras and their $\phi$-coordinated quasi modules, which were obtained before in a sequel, we construct a family of $\hbar$-adic quantum vertex algebras $V_L[[\hbar]]^{\eta}$ as deformations of the lattice vertex algebras $V_L$, and establish a natural connection between twisted quantum affine algebras of type $A, D, E$ and equivariant $\phi$-coordinated quasi modules for the $\hbar$-adic quantum vertex algebras $V_L[[\hbar]]^{\eta}$ with certain specialized $\eta$.

In this paper, we study a family of infinite-dimensional Lie algebras $\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where $\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$ naturally correspond to equivariant quasi modules for affine vertex algebras related to $\mathcal{L}_{S}$. Furthermore, for any finite cyclic group $S$, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types., Comment: 30 pages

In this paper, we give a unified construction of vertex algebras arising from infinite-dimensional Lie algebras, including the affine Kac-Moody algebras, Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds and deformations. We define a notion of what we call quasi vertex Lie algebra to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie algebra $\mathfrak{g}$, we construct a corresponding vertex Lie algebra ${\mathfrak{g}}_0$, and establish a canonical isomorphism between the category of restricted $\mathfrak{g}$-modules and that of equivariant $\phi$-coordinated quasi $V_{{\mathfrak{g}}_0}$-modules, where $V_{{\mathfrak{g}}_0}$ is the universal enveloping vertex algebra of ${\mathfrak{g}}_0$. This unified all the previous constructions of vertex algebras from infinite-dimensional Lie algebras and shed light on the way to associate vertex algebras with Lie algebras., Comment: 34 pages

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, 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 character $\chi$, 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., Comment: 33 pages

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., Comment: 31 pages

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 vertex 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.

This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-module nonlocal vertex algebra structure. We also give a construction of $\phi$-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.

In this paper, we study $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules. As an application, we also construct (equivariant) $\phi$-coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators.

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.

In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $\mu$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $\mu$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $\mu=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras., Comment: 66 pages. Final version

In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with $\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum 2-torus, the simple affine vertex algebra $L_{\widehat{sl_{\infty}}}(\ell,0)$ with $\ell$ a positive integer, and Levi subgroups $G$ of $GL_\ell(\mathbb{C})$. First, we give a canonical isomorphism between the category of integrable restricted $\widehat{sl_N}(\mathbb{C}_q)$-modules of level $\ell$ and that of equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Second, we classify irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Third, we establish a duality between irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules and irreducible regular $G$-modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-module. Fifth, we completely determine the following branchings: 1 The branching from $L_{\widehat{sl_{\infty}}}(\ell,0)\otimes L_{\widehat{sl_{\infty}}}(\ell',0)$ to $L_{\widehat{sl_{\infty}}}(\ell+\ell',0)$ for quasi modules. 2 The branching from $\widehat{sl_N}(\mathbb{C}_q)$ to its Levi subalgebras. 3 The branching from $\widehat{sl_N}(\mathbb{C}_q)$ to its subalgebras $\widehat{sl_N}(\mathbb{C}_q[t_0^{\pm M_0},t_1^{\pm M_1}])$., Comment: 54 Pages

Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $\mu$ a diagram automorphism of $\mathfrak{g}$ and $L(\mathfrak{g},\mu)$ the loop algebra of $\mathfrak{g}$ associated to $\mu$. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension $\widehat{\mathfrak{g}}[\mu]$ of $L(\mathfrak{g},\mu)$. The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras ([MRY]) as special examples. As an application, when $\mathfrak{g}$ is of simply-laced type, we prove that the classical limit of the $\mu$-twisted quantum affinization of the quantum Kac-Moody algebra associated to $\mathfrak{g}$ introduced in [CJKT1] is the universal enveloping algebra of $\widehat{\mathfrak{g}}[\mu]$.

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.

In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\mathcal{A}}}(\ell,0)$ (or $V_{\widehat{\mathcal{A}}}(2\ell,0)$). Furthermore, we determine irreducible modules and equivariant quasi modules for simple vertex algebra $L_{\widehat{\mathcal{A}}}(\ell,0)$ with $\ell$ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level $\ell$ for type $A$ trigonometric Lie algebra gives rise to an irreducible equivariant quasi $L_{\widehat{\mathcal{A}}}(\ell,0)$-module., Comment: 31 pages

Let $\preceq$ be a compatible total order on the additive group $\mathbb{Z}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4) \in \mathbb{C}^4$, we define $\mathbb{Z}^2$-graded Verma module $M(\mathbf{c}, \preceq)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c}, \preceq)$ to be irreducible is provided. Moreover, the maximal $\mathbb{Z}^2$-graded submodules of the Verma module $M(\mathbf{c}, \preceq)$ are characterized when $M(\mathbf{c}, \preceq)$ is reducible.

In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations., Comment: 15 pages

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 $\widetilde{\mathfrak g}$.

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 $\widetilde{L}$., Comment: 22 pages

Let $\fg$ be any untwisted affine Kac-Moody algebra, $\mu$ any fixed complex number, and $\wt\fg(\mu)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bm{\lambda}=({\lambda}_1, \cdots, {\lambda}_k)$ 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 $\wt V(\bm{\lambda},\bm{a})$ for the extended affine Lie algebra $\wt\fg(\mu)$. We prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is completely reducible. We also prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is integrable when all weights $\lambda_i$ in $\bm{\lambda}$ are dominant integral. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra $\wt\fg(\mu)$.

We establish a natural connection of the $q$-Virasoro algebra $D_{q}$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $\chi$, we define an infinite-dimensional Lie algebra $D_{S}$ which reduces to $D_{q}$ when $S=\mathbb{Z}$. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra ${\mathfrak{g}}_{S}$ with $S$ as an automorphism group and we prove that $D_{S}$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{{\mathfrak{g}}_{S}}$. We then relate restricted $D_{S}$-modules of level $\ell\in \mathbb{C}$ to equivariant quasi modules for the vertex algebra $V_{\widehat{\mathfrak{g}_{S}}}(\ell,0)$ associated to $\widehat{{\mathfrak{g}}_{S}}$ with level $\ell$. Furthermore, we show that if $S$ is a finite abelian group of order $2l+1$, $D_{S}$ is isomorphic to the affine Kac-Moody algebra of type $B^{(1)}_{l}$., Comment: 20 pages

In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$ which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family of associative algebras $\A(h)$. The associated quantum vertex algebras are based on the vacuum modules for $\A(h)$, whereas $\phi$-coordinated modules for these quantum vertex algebras are associated to $\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules., Comment: 33 pages. arXiv admin note: text overlap with arXiv:1505.07165

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-Nie's 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., Comment: 24 pages

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., Comment: 30 pages

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 $\psi:S_c\rightarrow L_r$ 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., Comment: 50 pages

A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method., Comment: 26 pages

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 affine 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 generating 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., Comment: latex, 25 pages

We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal Lie algebras., Comment: 36 pages

In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schr\"{o}dinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schr\"{o}dinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schr\"{o}dinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.

In this paper, the conjugate-linear anti-involutions and the unitary irreducible modules of the intermediate series over the twisted Heisenberg-Virasoro algebra are classified respectively. We prove that any unitary irreducible module of the intermediate series over the twisted Heisenberg-Virasoro algebra is of the form $\mathcal{A}_{a,b,c}$ for $a\in \mathbb{R}, b\in 1/2+\sqrt{-1}\mathbb{R}, c\in \mathbb{C}.$