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Abstract In this paper, we associate quantum vertex algebras to a certain family of associative algebras 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 ϕ -coordinated modules for these quantum vertex algebras are associated to A ˜ (g) -modules. Furthermore, we classify their irreducible ϕ -coordinated modules. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of infinite-dimensional Lie algebras X ˆ 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 X ˆ S with what are called the covariant algebras of the affine Lie algebra L S ˆ with respect to some automorphism groups, where L S is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted X ˆ S -modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to 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. [ABSTRACT FROM AUTHOR]

Abstract: 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 and of τ-modules using generating functions, where is proved to contain the evaluation modules while contains certain restricted τ-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories and . [Copyright &y& Elsevier]

Abstract: 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 we construct a vertex Leibniz algebra and show that can be embedded into a vertex algebra if and only if is a Lie algebra. [Copyright &y& Elsevier]

In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schrödinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schrödinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schrödinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

In this article, conjugate-linear anti-involutions and θ-unitary Harish-Chandra modules over the W-algebra W(2,2) are studied. It is proved that there are only two classes that conjugate-linear anti-involutions over W(2,2). The main result of this article is that a θ-unitary Harish-Chandra module over the W-algebra W(2,2) is simply a -unitary Harish-Chandra module over the Virasoro algebra and hence is either a simple highest or lowest weight module or a simple module from the intermediate series of the form A a,b for some [ABSTRACT FROM AUTHOR]

In this paper, Whittaker modules for the Schrödinger-Witt algebra sv are defined. The Whittaker vectors and the irreducibility of the Whittaker modules are studied. sv has a triangular decomposition according to its Cartan subalgebra h: sv=sv-+h+sv+. For any Lie algebra homomorphism ψ:sv+→C, we can define Whittaker modules of type ψ. When ψ is nonsingular, the Whittaker vectors, the irreducibility, and the classification of Whittaker modules are completely determined. When ψ is singular, by constructing some special Whittaker vectors, we find that the Whittaker modules are all reducible. Moreover, we get some more precise results for special ψ. [ABSTRACT FROM AUTHOR]

Abstract: We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras. [Copyright &y& Elsevier]

Let W be the Virasoro-like algebra, and d1, d2 be the degree derivations of W. Set L = W⊕C d1⊕C d2. Let Fα(V) be the L-module defined by Larsson's functor applied to a finite dimensional sl2-module V. In this article, the derivations from L to L-modules Fα(V) and the first cohomology group H1(L, Fα(V)) are given. [ABSTRACT FROM AUTHOR]

Abstract: Let be the q-analog Virasoro-like algebra. In this paper we prove that any Harish-Chandra module of with nontrivial center is isomorphic to a generalized highest weight module. Moreover, we show that a generalized highest weight Harish-Chandra module of is induced from an irreducible module of a Heisenberg subalgebra of . [Copyright &y& Elsevier]

To give a unified treatment on the association of Lie algebras and vertex algebras, we study (G , χ ϕ) -equivariant ϕ -coordinated quasi modules for vertex algebras, where G is a group with χ ϕ a linear character of G and ϕ is an associate of the one-dimensional additive formal group. The theory of (G , χ ϕ) -equivariant ϕ -coordinated quasi modules for nonlocal vertex algebra is established in [10]. 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 , χ ϕ) -equivariant ϕ -coordinated quasi modules. Furthermore, for any conformal algebra C , we construct a class of Lie algebras C ˆ ϕ [ G ] and prove that restricted C ˆ ϕ [ G ] -modules are exactly (G , χ ϕ) -equivariant ϕ -coordinated quasi modules for the universal enveloping vertex algebra of C. As an application, we determine the (G , χ ϕ) -equivariant ϕ -coordinated quasi modules for affine and Virasoro vertex algebras. [ABSTRACT FROM AUTHOR]

Let sl N ˆ (C q) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q. In this paper, we first prove that for any complex number ℓ , the category of restricted sl N ˆ (C q) -modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ (ℓ , 0) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ , the integrable restricted sl N ˆ (C q) -modules of level ℓ are exactly the twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify irreducible graded twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an infinite-dimensional Lie algebra D S for any abelian group S. If S is the additive group of integers, D S reduces to the q -Virasoro algebra D q introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra 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 g S ˆ. We then relate restricted D S -modules of level ℓ ∈ C to equivariant quasi modules for the vertex algebra V g S ˆ (ℓ , 0) associated to g S ˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q -Virasoro algebra D q and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2 l + 1 , D S is isomorphic to the affine Kac-Moody algebra of type B l (1). [ABSTRACT FROM AUTHOR]

Abstract Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g ˜ (μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k -tuple λ = (λ 1 , ⋯ , λ k) of weights of g , and k -tuple a = (a 1 , ⋯ , a k) of distinct non-zero complex numbers, we construct a class of modules V ˜ (λ , a) for the extended affine Lie algebra g ˜ (μ). We prove that the g ˜ (μ) -module V ˜ (λ , a) is completely reducible. We also prove that the g ˜ (μ) -module V ˜ (λ , a) is integrable when all weights λ i in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g ˜ (μ). [ABSTRACT FROM AUTHOR]

In this paper, we continue the study on toroidal vertex algebras initiated in [15] , to study concrete toroidal vertex algebras associated to toroidal Lie algebra L r ( g ˆ ) = g ˆ ⊗ L r , where g ˆ is an untwisted affine Lie algebra and L r = C [ t 1 ± 1 , … , t r ± 1 ] . We first construct an ( r + 1 ) -toroidal vertex algebra V ( T , 0 ) and show that the category of restricted L r ( g ˆ ) -modules is canonically isomorphic to that of V ( T , 0 ) -modules. Let c denote the standard central element of g ˆ and set S c = U ( L r ( 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 Z r -graded ring homomorphism ψ : S c → L r such that Im ψ is a Z r -graded simple S c -module. Denote by L ( ψ , 0 ) the simple quotient ( r + 1 ) -toroidal vertex algebra of V ( S c , 0 ) associated to ψ . We determine for which ψ , L ( ψ , 0 ) is an integrable L r ( g ˆ ) -module and we then classify irreducible L ( ψ , 0 ) -modules for such a ψ . For our need, we also obtain various general results. [ABSTRACT FROM AUTHOR]

In this paper, we study a certain deformation D of the Virasoro algebra that was introduced and called q -Virasoro algebra by Belov and Chaltikian, in the context of vertex algebras. Among the main results, we prove that for any complex number ℓ , the category of restricted D -modules of level ℓ 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 ℓ is canonically isomorphic to the category of Z -equivariant ϕ -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 gl ∞ . [ABSTRACT FROM AUTHOR]