Combinatorial Algebra

著名的群伯恩赛德问题在二十世纪初期首次提出，其问题是：每个有限生成的周期群都是有限的吗？ Kurosh-Levitzki 问题是一个关联代数类比问题：每个有限生成的 nil 代数都是有限维的吗？ 反例首先由 Golod 和 Shafarevich 在 20 世纪 60 年代提出。 因此，很自然地用额外的假设重新表述这些问题以获得正解。 1994 年，Zelmanov 因证明每个有限生成的残差有限群都是有限的而获得了著名的菲尔兹奖，从而解决了所谓的群的限制伯恩赛德问题。 我将泽尔曼诺夫在群论和李理论方面的开创性工作扩展到了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。 特别是，我已经证明，如果有限生成的零代数是无穷小 PI，则它是有限维的。 这是泽尔曼诺夫结果的重要概括。我在将我的新理论应用于卡普兰斯基问题方面也取得了一些成功，该问题解决了群代数的结构，其增广理想是雅各布森激进式。 泽尔马诺夫将卡普兰斯基问题称为群论中继受限伯恩赛德问题之后的下一个大障碍。

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.