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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]

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. [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]

Let Γ be a generic subgroup of the multiplicative group C ⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ , called twisted Γ -Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of Z N and an arbitrary finite order isometry of Z N preserving Q , we construct a family of twisted Γ -vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ -Lie algebras. As an 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. [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]

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 paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type A. [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]

The baby Tits-Kantor-Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac-Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp4(). [ABSTRACT FROM AUTHOR]

Let Derskew(q) be the skew derivation algebra of the quantum torus [image omitted] with q ≠ 1 a pth primitive root of unity. In this article, we investigate the automorphism group of the derived Lie subalgebra of the infinite dimensional Lie algebra [image omitted]. The result of the article generalizes those obtained in Xue et al. (2006) and Zheng and Tan (2007) with q = 1 or q being generic. [ABSTRACT FROM AUTHOR]

Abstract: Let be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible -module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight -modules are classified, and the intermediate series -modules are classified and constructed. [Copyright &y& Elsevier]

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998ab) and Kirkman et al. (1994). [ABSTRACT FROM AUTHOR]

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]

In this paper, we classify the irreducible integrable modules for nullity 2 toroidal extended affine Lie algebras with finite dimensional weight spaces. [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]

The article offers information on the 12th National Conference on Lie Algebras to be held by the Department of Mathematics at Huzhou Teachers College in China from July 20-26, 2011.

This is a paper in a series systematically to study toroidal vertex algebras and their (twisted) modules. 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 most of the extended affine Lie algebras can be associated to toroidal vertex algebras. [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 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]