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。

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.