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。

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.