Combinatorial Algebra

最早在二十世纪初提出的著名的群的伯恩赛德问题问：每个有限生成的周期群都是有限的吗？Kurosh-Levitzki问题是一个结合代数的类比：每个有限生成的零代数都是有限维的吗？反例首先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，很自然地用额外的假设来重新表述这些问题，以获得正解。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问题而获得了著名的菲尔兹奖。我已经把泽尔马诺夫在群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，它可以直接应用于代数的其他分支。特别地，我证明了有限生成的零代数是有限维的，如果它是无穷小的PI。这是对泽尔马诺夫研究结果的重要概括。我还成功地将我的新理论应用于Kaplansky的问题，该问题解决了其增广理想是Jacobson根的群代数的结构。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重大障碍。

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.