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This is a continuation of a previous study initiated by the third author 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 ϕ -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. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ be a diagram automorphism of g, and let L(g,μ) be the loop algebra of g associated to μ. 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] of L(g,μ). 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 is of simply-laced-type, we prove that the classical limit of the μ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g[μ]. [ABSTRACT FROM AUTHOR]

The conjugate-linear anti-involutions and 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 A a , b , c for a ∈ ℝ , b ∈ 1 2 + − 1 ℝ and c ∈ ℂ. [ABSTRACT FROM AUTHOR]

In this paper, we study what we call twisted ϕ -coordinated modules for nonlocal vertex algebras including weak quantum vertex algebras and we give a conceptual construction of nonlocal vertex algebras and their twisted ϕ -coordinated modules. As an illustrating example, we study certain vertex algebras and their twisted ϕ -coordinated modules, associated to certain infinite dimensional Heisenberg Lie algebras. As a reformulation and a slight extension of a result of Lepowsky, and a result of Doyon, Lepowsky and Milas, we also give a canonical connection between twisted modules for a general 1 T Z -graded nonlocal vertex algebra V and twisted ϕ -coordinated modules for the nonlocal vertex algebra obtained from V by Zhu's change-of-variable theorem. [ABSTRACT FROM AUTHOR]

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 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]

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: It is known that, up to isomorphism, there are only one lattice S and one non-lattice semi-lattice in . In [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342] we obtained a vertex operator representation for the baby-TKK algebra . The goal of the present paper is to study the TKK algebra , which includes the baby-TKK algebra as a subalgebra, and its vertex operator representations. As a by-product, we also obtain two representations for the baby-TKK algebra . One of the representations recovers the construction given in [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342]. [Copyright &y& Elsevier]

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]

Abstract: In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra. [Copyright &y& Elsevier]

Let Der(Cq) be the derivation Lie algebra of the quantum torus Cq. We construct a functor from gld-modules to Der(Cq)-modules, which generalizes the constructions obtained by Shen [Scientia Sinica A 29 (1986) 570], Larsson [Internat. J. Modern Phys. A 7 (1992) 6493], and Rao [J. Algebra 182 (1996) 401]. We also give a complete description of the structure of the Der(Cq)-modules when all the entries of the quantum torus matrix q=(qij)d×d are roots of unity. [Copyright &y& Elsevier]

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 A ˆ of Lie algebra A = gl ∞ ⊕ gl ∞ with respect to certain automorphism groups. We then prove that restricted modules of level ℓ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras V A ˆ (ℓ , 0) (or V A ˆ (2 ℓ , 0)). Furthermore, we determine irreducible modules and equivariant quasi modules for the simple vertex algebra L A ˆ (ℓ , 0) with ℓ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level ℓ for type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi L A ˆ (ℓ , 0) -module. [ABSTRACT FROM AUTHOR]

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 study (G , χ ϕ) -equivariant ϕ -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 , χ ϕ) -equivariant ϕ -coordinated quasi modules. As an application, we also construct (equivariant) ϕ -coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]