半群代数的范畴化、强nil-clean性和表示理论

Both semigroup algebras and the representation theory of algebras are vital research fields in algebra. The representation theory of semigroup algebras is an important branch of representation theory that has been widely used in topology, possibility and combinatorics. This subject first studies the semiprimitivity, ordinary quivers, representation types and hereditary of semigroup algebras, which contains the following parts: (1) by applying category algebras, locally Ehresmann semigroup algebras and orthodox semigroup algebras are studied; (2) we investigate the properties of locally inverse semigroup algebras by analyzing the structure of related sandwich matrices and constructing multiplicative subsemigroups; (3) since the idempotent sets of semigroups play an important role in the study of their semigroup algebras, we reduce the study of R-unipotent semigroup algebras to that of the semigroup algebras of left regular bands. Secondly, the strong nil-cleanness of completely 0-simple semigroup rings, the semigroup rings of the strong semilattice of semigroups and regular semigroup rings are characterized. Finally, we construct a correspondence between corkernal functors of hereditary algebras and a certain class of quotients of the related preprojective algebras; then use words in Coxeter groups to compute determiners of morphisms in hereditary algebras. This subject will enrich the categorization theory of semigroup algebras, provide a basis for combining the theory of semigroups and clean rings, and have a positive influence on the development of semigroup algebras and other related subjects.

半群代数和代数表示论都是代数学的重要研究内容。半群代数的表示理论作为表示论的一个重要研究分支，在拓扑、概率、组合等学科都有广泛的应用。本项目首先研究半群代数的半本原性、箭图、表示型和遗传性，包括以下几个方面：(1) 利用相关的范畴代数，研究局部Ehresmann半群代数和纯正半群代数；(2)通过分析相关夹心矩阵的结构和构建乘法子半群，研究局部逆半群代数；(3)基于半群的幂等元集在半群代数中的重要作用，把对R-幂幺半群代数的研究归结到左正则带代数。其次，研究完全0-单半群环、半群的强半格对应的半群环和正则半群环的强 nil-clean性。最后，建立遗传代数上的余核函子和对应的预投射代数上某类商模之间的对应关系；利用Coxeter群中字的长度计算遗传代数上同态的决定子。本项目将丰富半群代数的范畴化理论，为半群和clean环理论的结合提供依据，对半群代数和其它相关学科的发展有积极意义。

本项目主要研究了局部逆半群代数的π-半单性,R-unipotent的π-半单性,半群环的强nil-clean性等问题.具体内容如下.项目的第一部分,首先刻画一个半群何时以完全0-单半群为主因子.设R是一个含有单位元的交换环,且设S是一个幂等元集局部伪有限的局部逆半群.假设S 的主因子都是完全0-单半群.证明压缩半群代数R[S]是π-半单的当且仅当S的所有主因子所对应的半群代数是π-半单的.通过构造具体的例子证明局部伪有限条件对于上述结论是必要的.推广了关于局部逆半群代数的相关结果.项目的第二部分,设S是一个满足幂等元集E(S)局部L-有限的R-幂幺半群,构造了R[S]的一组正交幂等元集.然后证明B是R[S]的一组R-基,并且证明集合S=B∪{0}是R-幂幺完全0-单半群的0-直并.作为应用,证明了R[S]是π-半单的当且仅当S是一个逆半群且对于S的每一个极大子群G,群代数R[G]是π-半单的.项目的第三部分,研究了某些半群代数的强nil-clean 性.对于一个完全0-单半群M=M(G;I,Λ;P),证明了压缩半群代数R[M]是强nil-clean的当且仅当|I|=1或者|Λ|=1,且R[G]是强nil-clean的;作为推论,给出了局部逆半群代数的强nil-clean性的刻画.此外,设S是半群的强半格.则R[S]是强nil-clean的当且仅当对于每一个α∈Y,R[Sα]是强nil-clean的. 项目的第四部分,首先给出代数闭域上三维半群代数的幂等元集和Jacobson根,并且刻画了三维半群代数的同构类.通过计算箭图,研究了三维代数的表示型.进一步,证明一个三维(或者,二维)半群代数是胞腔的当且仅当它是交换的.作为推论,得到一个左零带所对应的半群代数是胞腔的当且仅当这个左零带是一个半格.