关于abel和幂零表代数的一些结果

【作者】 王志俊；

【导师】 海进科；

【作者基本信息】 青岛大学 ， 基础数学， 2006， 硕士

【摘要】 Arad和Blau在1991年从有限群的不可约特征标和共轭类的积的分解中抽象出表代数的概念,它是一类满足特定条件的定义在复数域上的有限维交换结合代数。表代数为研究上述两者的关系提供了一种统一的方法,并已成为一个相对独立的研究课题。 本论文从表代数的子结构和商结构的关系出发,对abel表代数和幂零表代数的结构进行了深入的探讨,并且得到一些有意义的结果。 Arad和Blau证明了abel表代数(A,B)等价于某个有限生成abel群G的群代数,受此启发本文第3节通过定义表基的一个群结构给出了abel表代数的结构定理,并从合成列数目的角度对初等abel表代数进行了细致刻画。 本文第4节对表代数的幂零性问题做了深入研究。我们知道满足中心扩张的群是幂零群。在给出幂零表代数基本性质的基础上,利用表代数的Jordan-Holder型定理我们证明了满足幂零扩张条件的表代数是幂零的,表代数的幂零性可以由表子集和商子集决定。同时会发现该条件在表代数幂零性的判别方面并不完全对应于幂零群的中心扩张。另外我们还将通过一例子说明该判别条件对abel表代数是不成立的。更多还原

【Abstract】 Arad and Blau abstracted the concept of table algebras, which are a class of finite dimensional, associative and commutative algebras over the complex numbers with certain specified properties, from the decompositions of products of either irreducible characters or conjugacy classes of finite groups. They provide a method in a uniform way to research the two cases mentioned above and have been a relatively independent researching subject.The present paper, from the point of view of the relationship between substructure and quotient structure, mainly characterizes structures of abelian and nilpotent table algebras and gains some meaningful results.Arad and Blau proved that an abelian table algebra (A, B) can be viewed as a group algebra of some abelian group G. Chapter 3 of this paper gives the structural theorem of abelian table algebras by defining a group structure in table basis. Furthermore, the structure of elementary abelian table algebras is discussed using the number of composition series of table algebras.Chapter 4 mainly deals with the nilpotency of table algebras. As is known, groups satisfying central extension conditions are nilpotent. we, firstly, obtain some basic properties on nilpotent table algebras, and then prove that table algebras which satisfy nilpotent extension are nilpotent by using Jordan-Holder type theorem of table algebras, i.e. nilpotency of a table algebra depends on that of their table subsets and quotient subsets, which is not fully corresponding to extension problems of nilpotent groups. Furthermore, we will find that conditions in the above theorem are not valid for abelian table algebras.更多还原