扭Heisenberg-Virasoro代数上的Poisson结构

Poisson代数是指同时具有代数结构和李代数结构的一类代数,其乘法与李代数乘法满足Leibniz法则.扭Heisenberg-Virasoro代数是一类重要的无限维李代数,是次数不超过1的微分算子李代数W(0)的普遍中心扩张,与曲线的模空间有密切联系.本文主要研究扭Heisenberg-Virasoro代数上的Poisson结构,首先确定了李代数W(0)上的Poisson结构,进而给出了扭Heisenberg-Virasoro代数上的Poisson结构.     

On Real Simple Left-symmetric Algebras

In this article, we commence to study the real (simple) left-symmetric algebras. From the known classification of certain complex (semi)simple left-symmetric algebras, we classify their corresponding real forms. We not only obtain the classification of real simple left-symmetric algebras in low dimensions, but also find certain examples of real simple left-symmetric algebras in higher dimensions. In particular, there exists a complex simple left-symmetric algebra without any real form. We also give a geometric construction for a class of real simple left-symmetric algebras. At last, we apply the classification results to study some structures related to geometry.     

扭Heisenberg-Virasoro代数上Hom-李代数结构

Hom-李代数是一类满足反对称和Hom-Jacobi等式的非结合代数.扭Heisenberg-Virasoro代数是次数不超过1的微分算子代数的中心扩张,它是一类重要的无限维李代数,与一些曲线的模空间有关.文章主要研究扭Heisenberg-Virasoro代数上Hom-李代数结构,确定了扭Heisenberg-Virasoro代数上存在非平凡的Hom-李代数结构.     

Generalized Heisenberg-Virasoro algebras

In this paper, we mainly study the generalized Heisenberg-Virasoro algebra. Some structural properties of the Lie algebra are obtained.      

Deformations on the Twisted Heisenberg-Virasoro Algebra

With the cohomology results on the Virasoro algebra, the authors determine the second cohomology group on the twisted Heisenberg-Virasoro algebra, which gives all deformations on the twisted Heisenberg-Virasoro algebra.     

Biderivations of the twisted Heisenberg–Virasoro algebra and their applications

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg–Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg–Virasoro algebra are given. It is also proved that every biderivation of the graded twisted Heisenberg–Virasoro left-symmetric algebra is trivial.     

A note on the left-symmetric algebraic structures of the Witt algebra

In Tang and Bai [Math Nachr. 2012;285:922–935] classified a class of non-graded left-symmetric algebraic structures on the Witt algebra under a certain rational condition. In this note, we show that this rational condition is not necessary. This leads to a more elegant classification of the left-symmetric algebraic structures and Novikov algebraic structures on the Witt algebra.     

P-twisted affine Lie algebra and its realizations by twisted vertex operators

In this paper, we define a P-twisted affine Lie algebra, and construct its realizations by twisted vertex operators.     

关于扭曲Lie结构范畴中的对偶函子

首先介绍扭曲Lie余代数．然后讨论扭曲Lie代数和扭曲Lie余代数范畴．主要给出扭曲Lie结构范畴之间函于的一些性质。     

Pointed representations of Virasoro algebra

In this paper we study the pointed representations of the Virasoro algebra. We show that unitary irreducible pointed representations of the Virasoro algebra are Harish-Chandra representations, thus they either are of highest or lowest weights or have all weight spaces of dimension 1. Further, we prove that unitary irreducible weight representations of Virasoro superalgebras are either of highest weights or of lowest weights, hence they are also Harish-Chandra representations. This work was supported by CNSF     

Smash product algebras over twisted dimodule algebras

Results of the research for smash product algebras over dimodule algebras are generalized to the more general twisted dimodule algebras.     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

On the action of Virasoro algebra on the space of univalent functions

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients.     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.