广义N=1，2超Virasoro代数的超双代数结构及表示
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摘要
众所周知,Virasoro代数作为一类无限维李代数,是线性微分算子(ti＋1d/(dt)|i∈Z)组成的无限维复李代数(Witt代数)的普遍中心扩张,其结构和表示在数学和理论物理的许多方面具有重要应用.在数学方面,Virasoro代数的表示对研究仿射Kac-Moody代数,moonshine模,顶点算子代数等都有重要作用.在理论物理方面,可参看Francesco, Mathieu和Senechal所著的共形场论[30]一书.超Virasoro代数及N=2超Virasoro代数可以分别看成是Virasoro代数的非平凡(?)2扩张和N=2超对称扩张,同样受到数学家和物理学家的广泛关注.
在量子群理论中,构造既非交换又非余交换的Hopf代数是一个核心问题.我们知道,从Yang-Baxter方程的解得出的量子群是统计量子力学中最令人感兴趣的.对李代数(?)而言,经典Yang-Baxter方程的解(r-矩阵)对应(?)上的一个“三角的”李双代数结构.因此,对带有“三角”结构的李双代数进行量子化就变得非常有意义.2000年,Ng和Taft证明了Virasoro代数上的李双代数结构都是三角上边缘的.本文的第一章从Virasoro代数的一个二维子代数出发,构造出Drinfel'd twist,利用Virasoro代数的普遍包络代数上的标准Hopf代数结构,得到一个新的非交换非余交换的Hopf代数.与传统的量子化不同,我们这里所用的二维子代数是由两个非局部有限元生成的.
量子超群是随着量子可逆散射理论推广到超系统[58]而自然出现的,李超双代数同样在其发展中占有重要地位.本文的第二章考虑超Virasoro代数(即N=1超Virasoro代数)及广义超Virasoro代数上的李超双代数结构.证明了超Virasoro代数上的李超双代数结构都是三角的.而对广义超Virasoro代数SVir[Γ, s]而言,因为其在SVir[Γ, s](?)SVir[Γ, s]上的一阶上同调未必是平凡的,所以它的李超双代数结构并不一定是三角的,我们给出了其成为三角的一个充分必要条件.
N=2超Virasoro代数是近些年由数学家和物理学家共同提出的一类代数.在本文的第三章,我们考虑N=2超Virasoro代数上的李超双代数结构,主要是Ramond N= 2超Virasoro代数(?),得到了其一阶上同调群是平凡的,从而证明了(?)上的李超双代数都是三角的.第四章我们对Topological N= 2超Virasoro代数的中间序列模进行了讨论.
The Virasoro algebra can be regarded as the universal central extension of the complex Lie algebra (Witt algebra) of the linear differential operators{ti+1d/(dt)|i∈Z). It is well known that the structure of Virasoro algebra together with their representations play important roles in many branches in both mathematics and theoretical physics. In mathematics, the representations of the Virasoro algebra have many important applications in the construction and the analysis of the struc-ture of affinc Kac-Moody algebras, the moonshine modules and the vertex operator algebras. A book on conformal field theory by Di Francesco, Mathieu and Senechal [30] gives a great detail on the connection between the Virasoro algebra and physics. The super-Virasoro and N= 2 super-Virasoro algebra are the nontrivial Z2-graded extension and N= 2 super-symmetric extension of the Virasoro algebra, respec-tively, which are also paid more attention by mathematician and physicist.
In the theory of quantum groups, it is a central problem to construct the non-commutative and noncocommutative Hopf algebra. We all know that the quantum groups obtained from the solutions of Yang-Baxter equation are very interesting. Let G be a Lie algebra, there is a one to one correspondence between "triangular" Lie bialgebra structures and the skew-symmetric solutions of the classical Yang-Baxter equation. Thus it is naturally to consider the quantizations of the triangular Lie bialgebras. In 2000, Ng and Taft proved that the Lie bialgebra structures on the Virasoro algebra are triangular coboundary. The thesis contains of four chapters. In chapter one, we consider the quantization of the Virasoro algebra W. From a non-abelian two dimensional Lie subalgebra of W, we construct a Drinfel'd twist and obtain a noncommutative and noncocommutative Hopf algebra by using the standard Hopf algebra structure on the universal enveloping algebra U(W). Differ- ent from the conventional methods of quantazation, the two-dimensional subalgbra is generated by two nonlocal finite elements.
Quantum supergroups appeared naturally when the quantum inversescattering method was generalized to the super-systems. Related Lie super-bialgebra are also important. In chapter two, we consider the Lie super-bialgebra structures on the super-Virasoro algebra (i.e., the N= 1 super-Virasoro algebra) and the generalized super-Virasoro algebra. It is proved that all Lie super-bialgebras on the super-Virasoro algebra are triangular. But to the generalized super-Virasoro algebra SVir[Γ, s], it is not the case because the first cohomology group of SVir[Γ, s] with coefficients in SVir[Γ, s](?)SVir[Γ, s] is not always trivial. A sufficient and necessary condition is given.
The N= 2 super-Virasoro algebra are obtained by mathematician and physi-cist. In chapter three, Lie bialgebra structures on the N= 2 super-Virasoro algebra are studied, especitively on the Ramond N= 2 super-Virasoro algebra, they are shown to be triangular by utilizing the first cohomology group. In chapter four, we discuss the modules of the intermediate series over Topological N= 2 super-Virasoro algebra.
在量子群理论中,构造既非交换又非余交换的Hopf代数是一个核心问题.我们知道,从Yang-Baxter方程的解得出的量子群是统计量子力学中最令人感兴趣的.对李代数(?)而言,经典Yang-Baxter方程的解(r-矩阵)对应(?)上的一个“三角的”李双代数结构.因此,对带有“三角”结构的李双代数进行量子化就变得非常有意义.2000年,Ng和Taft证明了Virasoro代数上的李双代数结构都是三角上边缘的.本文的第一章从Virasoro代数的一个二维子代数出发,构造出Drinfel'd twist,利用Virasoro代数的普遍包络代数上的标准Hopf代数结构,得到一个新的非交换非余交换的Hopf代数.与传统的量子化不同,我们这里所用的二维子代数是由两个非局部有限元生成的.
量子超群是随着量子可逆散射理论推广到超系统[58]而自然出现的,李超双代数同样在其发展中占有重要地位.本文的第二章考虑超Virasoro代数(即N=1超Virasoro代数)及广义超Virasoro代数上的李超双代数结构.证明了超Virasoro代数上的李超双代数结构都是三角的.而对广义超Virasoro代数SVir[Γ, s]而言,因为其在SVir[Γ, s](?)SVir[Γ, s]上的一阶上同调未必是平凡的,所以它的李超双代数结构并不一定是三角的,我们给出了其成为三角的一个充分必要条件.
N=2超Virasoro代数是近些年由数学家和物理学家共同提出的一类代数.在本文的第三章,我们考虑N=2超Virasoro代数上的李超双代数结构,主要是Ramond N= 2超Virasoro代数(?),得到了其一阶上同调群是平凡的,从而证明了(?)上的李超双代数都是三角的.第四章我们对Topological N= 2超Virasoro代数的中间序列模进行了讨论.
The Virasoro algebra can be regarded as the universal central extension of the complex Lie algebra (Witt algebra) of the linear differential operators{ti+1d/(dt)|i∈Z). It is well known that the structure of Virasoro algebra together with their representations play important roles in many branches in both mathematics and theoretical physics. In mathematics, the representations of the Virasoro algebra have many important applications in the construction and the analysis of the struc-ture of affinc Kac-Moody algebras, the moonshine modules and the vertex operator algebras. A book on conformal field theory by Di Francesco, Mathieu and Senechal [30] gives a great detail on the connection between the Virasoro algebra and physics. The super-Virasoro and N= 2 super-Virasoro algebra are the nontrivial Z2-graded extension and N= 2 super-symmetric extension of the Virasoro algebra, respec-tively, which are also paid more attention by mathematician and physicist.
In the theory of quantum groups, it is a central problem to construct the non-commutative and noncocommutative Hopf algebra. We all know that the quantum groups obtained from the solutions of Yang-Baxter equation are very interesting. Let G be a Lie algebra, there is a one to one correspondence between "triangular" Lie bialgebra structures and the skew-symmetric solutions of the classical Yang-Baxter equation. Thus it is naturally to consider the quantizations of the triangular Lie bialgebras. In 2000, Ng and Taft proved that the Lie bialgebra structures on the Virasoro algebra are triangular coboundary. The thesis contains of four chapters. In chapter one, we consider the quantization of the Virasoro algebra W. From a non-abelian two dimensional Lie subalgebra of W, we construct a Drinfel'd twist and obtain a noncommutative and noncocommutative Hopf algebra by using the standard Hopf algebra structure on the universal enveloping algebra U(W). Differ- ent from the conventional methods of quantazation, the two-dimensional subalgbra is generated by two nonlocal finite elements.
Quantum supergroups appeared naturally when the quantum inversescattering method was generalized to the super-systems. Related Lie super-bialgebra are also important. In chapter two, we consider the Lie super-bialgebra structures on the super-Virasoro algebra (i.e., the N= 1 super-Virasoro algebra) and the generalized super-Virasoro algebra. It is proved that all Lie super-bialgebras on the super-Virasoro algebra are triangular. But to the generalized super-Virasoro algebra SVir[Γ, s], it is not the case because the first cohomology group of SVir[Γ, s] with coefficients in SVir[Γ, s](?)SVir[Γ, s] is not always trivial. A sufficient and necessary condition is given.
The N= 2 super-Virasoro algebra are obtained by mathematician and physi-cist. In chapter three, Lie bialgebra structures on the N= 2 super-Virasoro algebra are studied, especitively on the Ramond N= 2 super-Virasoro algebra, they are shown to be triangular by utilizing the first cohomology group. In chapter four, we discuss the modules of the intermediate series over Topological N= 2 super-Virasoro algebra.
引文
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