正则半群代数的理想链

The Chain of Ideals of Regular Semigroup Algebras

摘要: 胞腔代数的出现完满地解答了表示论中的一个最基本的问题——确定不可约表示的参数集。Graham和Lehrer利用胞腔基给出了胞腔代数的定义，König和Xi则是利用胞腔理想链给出了胞腔代数的等价定义。通过胞腔理想链的定义，可以更好地研究胞腔代数的结构和同调性质。由于正则半群是一类重要的半群，它构成了半群代数理论的主要研究领域之一。因此本文将从胞腔代数的胞腔理想链出发，去研究正则半群代数的双边理想链。本文的主要结果是若正则半群代数具有一条胞腔理想链时，则某些极大子群的群代数都具有一条双边理想链，相反，当某些极大子群的群代数都具有一条胞腔理想链时，正则半群代数将会具有一条与其相关的双边理想链。由于胞腔理想链也是双边理想链，因此研究代数的双边理想链对研究代数的胞腔性是很有帮助的。

Abstract: The emergence of cellular algebras has solved one of the most basic problems in representation theory that determines parameter set of irreducible representation. Graham and Lehrer gave the definition of cellular algebras by using cellular basis; König and Xi gave the equivalent definition of cellular algebras by using the chain of cellular ideals. The structure and homology properties of cellular algebras can be better studied through the definition of the chain of cellular ideals. Regular semigroup is an important kind of semigroup, which constitutes one of the main research fields of the theory of semigroup algebras. In this paper we will study the chains of two-sided ideals of regular semigroup algebras from the chains of cellular ideals of cellular algebras. The main result of this paper is that if the semigroup algebra of a regular semigroup has the chain of cellular ideals, the group algebras of certain maximal subgroups all have the chains of two-sided ideals, and conversely, when the group algebras of certain maximal subgroups all have the chains of cellular ideals, the semigroup algebra of a regular semigroup will have the chain of two-sided ideals associated with them. Since the chains of cellular ideals are also the chains of two-sided ideals, it is helpful to study the chains of two-sided ideals of algebras to study the cellularity of algebras.