Combinatorial algebra: identities, actions and gradings

伯恩赛德问题最早提出于20世纪初，当时群的问题是：每个有限生成的周期群都是有限的吗？Kurosh-Levitzki问题是一个代数类比：每个有限生成的代数代数都是有限维的吗？反例最早是由戈洛德和沙法雷维奇在20世纪60年代的S构造的。因此，为了获得正解，用额外的假设重新表述这些问题是很自然的。卡普兰斯基或多或少地发明了多项式单位代数领域，以便给出这个问题的最佳可能的解决方案。1994年，Zelmanov因证明了每个有限生成的剩余有限群是有限的，从而解决了群的所谓受限Burnside问题而获得了菲尔兹奖。为此，他首先给出了某类李代数的Kurosh-Levtzki问题的一个正解。这些令人惊叹的结果，加上为证明它们而开发的强大理论，启发了我25年多来的研究计划。研究所谓的伯恩赛德类型的问题一直是我研究计划的核心。这类问题自然而然地出现在代数的所有领域：群论和结合代数，以及非结合代数，如李代数和乔丹代数。其想法是从表面上看是微弱的局部条件来推断全球现象。例如，我研究了在只知道一个更小、更弱的关系集合一次只存在于几个元素之间的情况下，什么时候可以推导出全球定律--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。第一个主题讨论结合代数和李代数的多项式恒等式以及给定的“亚纯”作用。亚纯作用包括群分次作用、Hopf代数作用、对合作用、导子作用和反导子作用以及代数自身的左正则作用。我试图推广和统一PI理论的一系列关键结果，证明了如果这种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。事实上，我猜想，一个人只需要涉及有限长度的动作的关系就可以得出这样的结果。*我的提议的第二个主题解决了代数的动词子空间。“动词”子空间是由代数中的一个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外，这是非对易PI-理论的一个全新的研究领域。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语言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. *****