Combinatorial algebra: identities, actions and gradings

伯恩赛德问题最早提出于世纪，它提出了这样一个问题：是否每个周期群都是有限的？Kurosh-Levitzki问题是一个代数类似问题：是否每个生成的代数代数都是有限维的？反例最早是由Golod和Shafarevich在20世纪60年代提出的。因此，很自然地，为了获得积极的解决办法，需要用额外的假设重新表述这些问题。Kaplansky或多或少地发明了多项式恒等代数领域，以给出他对这个问题的最佳解决方案。Zelmanov赢得了菲尔兹奖于1994年，他的证明，每一个生成的剩余有限群是有限的，从而解决了所谓的限制伯恩赛德问题的群体。为了做到这一点，他首先给出了一个积极的解决方案Kurosh-Levtzki问题的李代数的某种类型。这些惊人的结果，以及为了证明它们而发展的强大理论，激励了我自己超过25年的研究计划。研究所谓的伯恩赛德型问题一直是我研究计划的核心。这类问题在代数的所有领域都很自然地出现：群论和结合代数，以及非结合代数，如李代数和约当代数。这个想法是从表面上看起来是弱局部条件的东西推导出全局现象。例如，我研究了当一个代数只知道一个更小的、更弱的关系集合在一次取的几个元素之间成立时，什么时候可以推导出全局定律--即多项式恒等式--在代数中成立。我目前的建议有两个关键主题。第一个主题讨论了具有给定“亚纯”作用的结合代数和李代数的多项式恒等式。亚纯作用包括群的分次、霍普夫代数作用、对合作用、导子和反导子作用以及代数在自身上的左正则作用。 我试图通过证明如果这样的代数满足一个涉及动作的恒等式，那么它实际上满足一个普通的多项式恒等式，来扩展和统一PI理论的一系列关键结果。事实上，我猜想，为了得出这样一个结果，人们只需要涉及有界长度作用的关系。我的建议的第二个主题涉及代数的动词子空间。一个“动词”子空间是由代数中多项式的所有值生成的子空间。除了几个非常特殊的多项式，这是一个全新的领域的调查非交换PI理论。首先考虑的问题之一是：如果动词子空间是有限维的，那么由动词子空间生成的动词子代数和动词理想也是有限维的吗？每个动词子空间都有一个有趣的对偶：由多项式的“零”生成的“边缘”子空间。*****

First posed early in the 20th century, the Burnside Problem for groups asked: is every finitely generated periodic group finite? The Kurosh-Levitzki Problem is an algebraic analogue: is every finitely generated algebraic algebra finite-dimensional? Counterexamples were first constructed by Golod and Shafarevich in the 1960's. It was natural, therefore, to reformulate these problems with additional hypotheses in order to obtain positive solutions. Kaplansky more-or-less invented the field of polynomial identity algebras in order to give his best possible solution to this problem. Zelmanov won the Fields Medal in 1994 for his proof that every finitely generated residually finite group is finite, thereby solving the so-called Restricted Burnside Problem for groups. In order to do this, he gave first a positive solution to the Kurosh-Levtzki Problem for Lie algebras of a certain type. These amazing results, together with the powerful theory developed in order to prove them, have inspired my own research program for over twenty-five years.******Studying problems of so-called Burnside-type has always been at the core of my research program. These sorts of problems occur naturally in all areas of algebra: group theory and associative algebra, as well as nonassociative algebra, like Lie algebra and Jordan algebra. The idea is to deduce global phenomena from what appears on the surface to be weak local conditions. For example, I investigate when global laws - namely, polynomial identities - can be deduced to hold in an algebra knowing only that a smaller, weaker, collection of relations hold among a few elements taken at a time.******My current Proposal has two key themes. The first theme addresses the polynomial identities of associative and Lie algebras with a given "hypomorphic" action. Hypomorphic actions include gradings by groups, Hopf algebra actions, actions by involutions, actions by derivations and anti-derivations, and the left regular action of an algebra on itself. I seek to extend and unify a series key results from PI-theory by proving that if such algebras satisfy an identity involving the action, then it actually satisfies an ordinary polynomial identity. In fact, I conjecture that one only needs relations involving the action of bounded length in order to conclude such a result.******The second theme of my Proposal addresses the verbal subspaces of an algebra. A "verbal" subspace is the subspace generated by all the values a polynomial in an algebra. Apart from a few very special polynomials, this is a completely new area of investigation in noncommutative PI-theory. One of the first problems under consideration asks: if the verbal subspace is finite-dimensional, does it follow that the verbal subalgebra and verbal ideal generated by the verbal subspace is also finite-dimensional? There is an interesting dual to every verbal subspace: the "marginal" subspace generated by the "zeros" of a polynomial. *****