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Let g be a (twisted or untwisted) affine Kac-Moody algebra, and μbe a diagram automorphism of g. In this paper, we give an explicit realization for the universal central extension ĝ[μ] of the twisted loop algebra of g with respect to μ, which provides a Moody-Rao-Yokonuma presentation for the algebra ĝ[μ] when μ, is non-transitive, and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.

Let ⪯ be a compatible total order on the additive group ℤ2, and Lbe the rank two Heisenberg-Virasoro algebra. For any c= (c1, c2, c3, c4) ∈ℂ4, we define a ℤ2-graded Verma module M(c, ⪯) for L. A necessary and sufficient condition for M(c, ⪯) to be irreducible is provided. Moreover, the maximal ℤ2-graded submodules of M(c, ⪯) are characterized when M(c, ⪯) is reducible.

In this paper, we give explicit realizations for the irreducible integrable modules, which were classified in Chang and Tan [Pacific J Math, 2011, 252: 293–312], of the extended baby TKK algebra. Moreover, conditions for these modules to be unitary are determined.In this paper, we give explicit realizations for the irreducible integrable modules, which were classified in Chang and Tan [Pacific J Math, 2011, 252: 293–312], of the extended baby TKK algebra. Moreover, conditions for these modules to be unitary are determined.

Abstract: Let DerA be the Lie algebra of derivations of the d-torus $$ A = \mathbb{C}[t_1^{ \pm 1} ,...,t_d^{ \pm 1} ] $$. By applying Shen-Larsson’s functors we get a class of indecomposable DerA-modules from finite-dimensional indecomposable gl d -modules. We also give a complete description of the submodules of these indecomposable DerA-modules. Our results generalize those obtained by Rao.

In this paper we study the representation theory for certain "half lattice vertex algebras." In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.

Let T(S) be the Jordan algebra constructed from a semilatticeSof Rν(ν≥1) [see Allisonet al., Mem. Amer. Math. Soc.,603(1997)]. Let G(T(S)) be the Lie algebra obtained from the Jordan algebra T(S) by the Tits–Kantor–Koecher construction. The TKK algebraG^(T(S)) is defined to be the universal central extension of the Lie algebra G(T(S)). In this paper we present a complete description of the TKK algebraG^(T(S)), whereSis the smallest possible (nonlattice) semilattice, which then allows us to give a representation to this Lie algebra by vertex operators.

Vertex operator representations for toroidal Lie algebras of simply-laced type were studied in [MRY] and [EM]. In this paper we give vertex operator constructions for the toroidal Lie algebra of type Bl (l ≥ 3). The constructions are related to the folding of Dynkin diagram of [image omitted]  and a two-cocycle necessary for the vertex operator constructions.

The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrödinger equations with magnetic effect. Under certain conditions, we prove the existence and nonexistence of the non-trivial free asymptotic solutions. In addition, the decay estimates of the solutions are also obtained.

In this paper we consider the semi-discretization difference method for the system of Zakharov equations. Under certain conditions, the convergence, error stimates and stability of the given difference scheme are studied.