Combinatorial algebra: identities, actions and gradings

伯恩赛德问题（Burnside Problem）在20世纪初首次提出，它的问题是：是否每个有限生成的周期群都是有限的？Kurosh-Levitzki问题是一个代数类比：是否每一个有限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，很自然地要用额外的假设来重新表述这些问题，以便得到正解。卡普兰斯基或多或少发明了多项式恒等代数领域，就是为了给出这个问题的最佳可能解。Zelmanov在1994年获得了菲尔兹奖，因为他证明了每一个有限生成的剩余有限群都是有限的，从而解决了所谓的群的限制性Burnside问题。为了做到这一点，他首先给出了某一类李代数的Kurosh-Levtzki问题的正解。这些惊人的结果，以及为了证明它们而发展起来的强有力的理论，激励了我自己超过25年的研究计划。******研究所谓burnside类型的问题一直是我研究计划的核心。这类问题自然出现在代数的各个领域：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微弱的局部条件推断出全球现象。例如，我研究了全局定律——即多项式恒等式——何时可以被推导为在代数中成立，只知道在一次取的几个元素之间存在一个较小的、较弱的关系集合。******我目前的提案有两个关键主题。第一个主题讨论具有给定“半胚”作用的结合子代数和李代数的多项式恒等式。次纯作用包括群作用、Hopf代数作用、对合作用、导和反导作用以及代数对自身的左正则作用。我试图通过证明如果这样的代数满足一个涉及作用的恒等式，那么它实际上满足一个普通多项式恒等式，来扩展和统一pi理论的一系列键结果。事实上，我推测，为了得出这样的结论，人们只需要涉及有界长度作用的关系。******我的提案的第二个主题是代数的语言子空间。“动词”子空间是由代数中多项式的所有值生成的子空间。除了一些非常特殊的多项式外，这是非交换pi理论中一个全新的研究领域。首先要考虑的问题之一是：如果语言子空间是有限维的，那么由语言子空间生成的语言子代数和语言理想是否也是有限维的？每个文字子空间都有一个有趣的对偶：由多项式的“零”生成的“边际”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. *****