Identities in Algebras and Pro-P Groups

数系所满足的一个基本恒等式是交换律ab=ba。更复杂的代数系统，如函数集或对称可以有更复杂的行为，反映更复杂的代数恒等式。这个研究项目解决的问题是，所有这些恒等式的集合是否可以从一个有限的子集中推导出来。研究结果有望在代数、数论和理论计算机科学中得到应用。本课题的目的是发展一个新的无限群理论，满足亲单能或亲p恒等式。该理论应与具有多项式恒等式的代数理论平行，并为研究交换环上的线性群提供一种新的途径。实现这个程序的重要步骤将是(1)证明线性的亲p群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的，(3)一个结构理论，它将建立在某些条件下具有恒等式的群是线性的。

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.