Identities in Algebras and Pro-P Groups

代数和 Pro-P 群中的恒等式

• 批准号: 1601920
• 负责人: Efim Zelmanov
• 金额: $ 19.92万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601920&HistoricalAwards=false
• 关键词: Identities; Algebras; Pro; Groups

One of the fundamental identities satisfied by number systems is the commutative law ab=ba. More complex algebraic systems such as sets of functions or symmetries can have much more intricate behavior, reflected by more complicated algebraic identities. This research project addresses questions such as whether the set of all such identities can be deduced from a finite subset. The results are expected to have applications in algebra, number theory, and theoretical computer science. This project aims to develop a new theory of infinite groups satisfying pro-unipotent or pro-p identities. The theory should be parallel to the theory of algebras with polynomial identities and provide a new approach to linear groups over commutative rings. Important steps in implementation of this program will be (1) a proof that linear pro-p and pro-unipotent groups satisfy nontrivial identities, (2) proofs that these identities are finitely based, and (3) a structure theory that would establish that under certain conditions groups with identities are linear.

数系满足的基本恒等式之一是交换律 ab=ba。更复杂的代数系统，例如函数集或对称性，可以具有更复杂的行为，通过更复杂的代数恒等式反映。该研究项目解决的问题包括是否可以从有限子集中推导出所有此类身份的集合。研究结果预计将在代数、数论和理论计算机科学中得到应用。该项目旨在发展一种满足亲单能或亲 p 恒等式的无限群的新理论。该理论应该与具有多项式恒等式的代数理论平行，并为交换环上的线性群提供一种新的方法。实施该计划的重要步骤将是（1）证明线性亲p和亲单能群满足非平凡恒等式，（2）证明这些恒等式是有限基础的，以及（3）结构理论，该理论将建立在某些条件下具有恒等式的群是线性的。