Quotient Structures in C-Algebras
Quotient Structures in C-Algebras
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DOI：
10.1006/jabr.1995.1174
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发表时间：
1995-07
影响因子：
0.9
通讯作者：
H. I. Blau
中科院分区：
文献类型：
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作者：
H. I. Blau
一个C-代数是复数上的交换代数A，它有一个特殊的基B，它具有某些特定的性质（下面给出）。特别地，结构常数是真实的。C-代数的定义首先是由Kawada [K]在Hoheisel [H]之后明确提出的，尽管这个概念在Schur以前的工作中是隐含的（[S1]或[S2，Vol. III，pp. 266-291]）。川田的目标是抽象之间的关系中心的组代数和环类功能的有限组。因此，对于任何C-代数（A，B），他构造了一个对偶C-代数（A，B），使得B= B，并且使得当A是一个群代数的中心，其基B由共轭类上的和组成时，则（A，B）可以等同于以群的不可约特征标集为基的类函数代数如果结构常数非负，则（A，B）是表代数（如Arad和作者[AB]所定义）。Kawada为C-代数定义了合适的子结构和商结构的概念。在（A，B）和（A，B）都是表代数的前提下，他证明了（A，B）的任何子结构都产生商结构，反之亦然。这些假设成立，当然，只要C-代数产生于有限群，如在上面的段落中，或当它是一个交换结合方案的邻接代数（见[BI，第IL 2，II。3];这里引用的川田的结果，由McMullen和Price [M，MP]以“有限交换超群”的名义独立地重新发现，在[B1，第II节]中给出。5、二。9]）。但有C-代数的例子（见考试-
A C-algebra is a commutative algebra A over the complex numbers with a distinguished basis B which has certain specified properties (given below). In particular, the structure constants are real. The definition of a C-algebra was first stated explicitly by Kawada [K], following Hoheisel [H], although the notion is implicit in previous work of Schur ([S1] or [S2, Vol. III, pp. 266—291]). Kawada’s goal was to abstract the relationship between the center of the group algebra and the ring of class functions for a finite group. Thus, fqI_r any C-algebra (A, B), he constructed a dual C-algebra (A, B), so that B= B, and so that when A is the center of a group algebra with basis B consisting of the sums over conjugacy classes, then (A, B) may be identified with the algebra of class functions with basis the set of irreducible characters of the group (each multiplied by its degree; see below).If the structure constants are nonnegative, then (A, B) is a table algebra (as defined by Arad and the author [AB]). Kawada defined suitable notions of substructure and quotient structure for C-algebras. Under the assump-tion that both (A, B) and (A, B) are table algebras, he showed that any substructure of (A, B) yields a quotient structure, and conversely. These assumptions hold, of course whenever the C-algebra arises from a finite group as in the paragraph above, or when it is the adjacency algebra of a commutative association scheme (see [BI, Sections IL2, II. 3]; Kawada’s results cited here, which were independently rediscovered in the guise of “finite abelian hypergroups” by McMullen and Price [M, MP], are given in [B1, Sections II. 5, II. 9]). But there are examples of C-algebras (see Exam-