Combinatorial Algebra

在二十世纪早期，著名的伯恩赛德问题（Burnside Problem）首次提出：是否每一个有限生成的周期群都是有限的？Kurosh-Levitzki问题是一个关联代数的类比：是否每一个有限生成的零代数都是有限维的？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，以便得到一个正解。Zelmanov在1994年获得了著名的菲尔兹奖，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。我已经将Zelmanov在群论和李论方面的开创性工作扩展到一个单一的统一理论，该理论可以直接应用于代数的其他分支。特别地，我已经证明了一个有限生成的零代数是有限维的，如果它是无穷小PI。这是对Zelmanov结果的一个重要推广。我还成功地将我的新理论应用于卡普兰斯基问题，该问题解决了群代数的结构，其增广理想是Jacobson根。Zelmanov将Kaplansky问题称为继限制性Burnside问题之后群论的下一个大障碍。

First posed early in the twentieth century, the famous Burnside Problem for groups asked: Is every finitely generated periodic group finite? The Kurosh-Levitzki Problem is an associative algebra analogue: Is every finitely generated nil 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 a positive solution. Zelmanov won the prestigious 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. I have extended Zelmanov's seminal work in group and Lie theory to a single unified theory that has direct applications to other branches of algebra. In particular, I have proved that a finitely generated nil algebra is finite-dimensional if it is infinitesimially PI. This is a significant generalisation Zelmanov's results. I have also had some success in applying my new theory to Kaplansky's Problem which addresses the structure of group algebras whose augmentational ideal is Jacobson radical. Zelmanov has referred to Kaplansky's Problem as the next big hurdle in group theory after the Restricted Burnside Problem.