Mathematical Sciences: Minimal Embeddings in Central Simple Algebras

本研究项目涉及环理论中的问题，更确切地说，涉及半简单代数在中心简单代数中的嵌入问题。如果A是一个半简单K-代数，其中K是一个全局域，那么本研究的一个目标是确定一个中心简单K-代数的最小维数，其中A嵌入到D上的t x t矩阵中，其中D是一个K-除环。我们将考虑的一个相关问题是，对于给定的群G，确定最小t，使得D上的t x t矩阵是G对某个k分环D的交叉积。本项目所支持的研究属于环理论的一般领域。具体来说，首席研究员将研究一个中心简单代数如何包含另一个中心简单代数。这项工作借鉴了群论的一些深刻结果，并将阐明代数和数论中的各种问题。

This research project is concerned with problems in ring theory and deals, more precisely, with questions about the embedding of semi-simple algebras in central simple algebras. If A is a semi-simple K-algebra where K is a global field, then one goal of this investigation is to determine the minimum dimension of a central simple K-algebra in which A embeds and the minimum t such that A embeds in the t x t matrices over D, where D is a K- division ring. A related problem, which will be considered, is to determine for a given group G the minimum t such that the t x t matrices over D is a crossed product for G for some K-division ring D. The research supported in this project is in the general area of ring theory. Specifically, the principal investigator will study how one central simple algebra can contain another. This work draws on some deep results in group theory and will shed light on various questions in algebra and number theory.