论文标题
模块化代表理论和交换Banach代数
Modular representation theory and commutative Banach algebras
论文作者
论文摘要
在最近的Benson和Symonds的一篇论文中，引入了一个新的不变式，以用于有限群体的模块化表示。相对于表示环的Banach代数完成，解释为光谱半径。我们的目的是进一步采取这些概念，并研究由此产生的Banach代数的结构。该论文中的某些材料在此处重复了更大的一般性，并为了清楚说明。 我们给出了一个抽象表示环的公理定义，并给出了理想的表示。然后，完成是一个交换性的Banach代数，并应用了1940年代的Gelfand技术，以研究代数同构的空间至$ \ MATHBB C $。这项研究的一个令人惊讶的结果是，雅各布森自由基和（复杂的）表示环的零自由基总是重合的。 这些注释旨在代表理论家。因此，详细给出了交换Banach代数的背景材料，而表示理论背景更加凝结。
In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to $\mathbb C$. One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.
