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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]

Let U ħ (g ^) be the untwisted affinization of a symmetrizable quantum Kac-Moody algebra U ħ (g) . For ℓ ∈ C , we construct an ħ -adic quantum vertex algebra V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between ϕ -coordinated V g ^ , ħ (ℓ , 0) -modules and restricted U ħ (g ^) -modules of level ℓ . Suppose that ℓ is a positive integer. We construct a quotient ħ -adic quantum vertex algebra L g ^ , ħ (ℓ , 0) of V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between certain ϕ -coordinated L g ^ , ħ (ℓ , 0) -modules and restricted integrable U ħ (g ^) -modules of level ℓ . Suppose further that g is of finite type. We prove that L g ^ , ħ (ℓ , 0) / ħ L g ^ , ħ (ℓ , 0) is isomorphic to the simple affine vertex algebra L g ^ (ℓ , 0) . [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]

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]