【作者】 夏春光；

【导师】 苏育才； 张瑞斌；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2013， 博士

【摘要】 在本论文中,我们研究了一类无限维Z2-分次但非有限Z-分次的Block型李代数B(q)的结构理论和表示理论,其中q是一个非零复参数.第一章介绍了研究背景以及本论文的主要结果.Block型李代数最早由Richard Earl Block [7]在五十年代末期引入,这类李代数可以看成是Zassenhaus代数在特征零情形下的类比.李代数B(q)与著名的Virasoro代数和W1+∞代数都有着紧密的联系.李代数B(q)还可以看成是特殊的(广义)Cartan型李代数.众所周知Virasoro代数和W1+∞代数在共形场论,量子霍尔效应等各种物理理论中起着非常重要的作用,Cartan型李代数的表示理论也远未发展完善.因而研究B(q)的结构理论和表示理论非常有意义.第二章研究了B(q)的结构理论.假设参数q是正整数.通过研究B(q)的局部有限元和局部幂零元,我们首先刻画了召(q)的自同构群,并给出了召(q)的同构分类.由于B(q)是非有限生成Z-分次李代数,我们利用文献[61]中的技巧刻画了B(q)的导子代数.另外,我们还统一刻画了B(q)的中心扩张.最后,我们讨论了当参数q不是正整数时B(q)的结构理论.第三章研究了B(q)的表示理论.由于B(q)含有Virasoro子代数,受Olivier Mathieu[39]的关于Virasoro代数Harish-Chandra模分类结果的启发,这一章我们大致分类了B(q)的拟有限不可约模.一般的无限维分次李代数拟有限模的概念是由Victor Kac和Andrey Radul[29]在研究W1+∞代数的表示理论时首次提出的.Kac等人在[6,17,29,31]中指出：这种无限维非有限Z-分次李代数的拟有限表示是个非常不平凡的问题.另外,我们还分类了B(q)的拟有限不可约最高权模和不可约中间序列模.第四章继续研究B(q)的表示理论.在第三章关于召(q)的拟有限不可约最高权模分类的基础上,结合Virasoro代数最高权酉模的分类结果,我们完全分类了B(q)的拟有限不可约最高权酉模.这一分类结果表明,B(q)的拟有限不可约最高权酉模几乎可以视为Virasoro代数不可约最高权酉模的单参数推广.更多还原

【Abstract】 In this thesis, we study the structure theory and representation theory of a class of infinite-dimensional Z2-graded but not-finitely Z-graded Block type Lie algebras B(q), where q is a nonzero complex parameter.In Chapter1, we introduce the research background and the main results of this thesis. Block type Lie algebras were firstly introduced by Richard Earl Block [7] in the late50s, which can be viewed as analogous of the Zassenhaus algebras in characteristic zero case. There exist intimate relations between the Lie algebras B(q) and the well-known Virasoro algebra and W1+∞algebra. The Lie algebra B(q) can be also viewed as some special cases of (generalized) Cartan-type Lie algebras. It is well-known that the Virasoro algebra and W1+∞algebra play very important roles in various physical theories, such as conformal field theory, the theory of the quantum Hall effect, etc. The representation theory of Cartan type Lie algebras is far from being well developed. So, it is very interesting to study the structure theory and representation theory of B(q).In Chapter2, we study the structure theory of B(q). Suppose that the parameter q are positive integers. By studying the locally finite elements and locally nilpotent elements of B(q), we firstly characterize the automorphism group of B(q) and give the isomorphic classification of B(q). Since B(q) is not finitely-generated Z-graded Lie algebra, we employ a technique developed in [61] to characterize the structure of the derivation algebra of B(q). In addition, we also uniformly characterize the central extensions of B(q). At last, we discuss the structure theory of B(q) when the parameter q are not positive integers.In Chapter3, we study the representation theory of B(q). Since B(q) contains a Virasoro subalgebra, motivated by Olivier Mathieu’s classification on Harish-Chandra modules over the Virasoro algebra [39], we give a rough classification of the quasifinite irreducible modules of B(q) in this chapter. The notion of general quasifinite modules over infinite dimensional graded Lie algebras was firstly proposed by Victor Kac and Andrey Radul [29] when studying the representation theory of the W1+∞algebra. Kac et al.[6,17,29,31] pointed out that the quasifinite representation of this kind of infinite-dimensional not-finitely Z-graded Lie algebras is a highly nontrivial problem. In addition, we also classify the quasifinite irreducible highest weight B(g)-modules and the irreducible B(q)-modules of the intermediate series.In Chapter4, we continue to study the representation theory of B(q). Based on the classification of the quasifinite irreducible highest weight B(q)-modules in Chapter3, together with the classification of the highest weight unitary modules over Vira-soro algebra, we completely classify the quasifinite irreducible highest weight unitary modules over B(q). This classification indicates that the quasifinite irreducible highest weight unitary modules over B(q) can be almost viewed as a single-parameter gener-alization of the irreducible highest weight unitary modules over Virasoro algebra.更多还原