Combinatorial algebra: identities, actions and gradings

点击翻译按钮获取中文摘要

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.******Studying problems of so-called Burnside-type has always been at the core of my research program. These sorts of problems occur naturally in all areas of algebra: group theory and associative algebra, as well as nonassociative algebra, like Lie algebra and Jordan algebra. The idea is to deduce global phenomena from what appears on the surface to be weak local conditions. For example, I investigate when global laws - namely, polynomial identities - can be deduced to hold in an algebra knowing only that a smaller, weaker, collection of relations hold among a few elements taken at a time.******My current Proposal has two key themes. The first theme addresses the polynomial identities of associative and Lie algebras with a given "hypomorphic" action. Hypomorphic actions include gradings by groups, Hopf algebra actions, actions by involutions, actions by derivations and anti-derivations, and the left regular action of an algebra on itself. I seek to extend and unify a series key results from PI-theory by proving that if such algebras satisfy an identity involving the action, then it actually satisfies an ordinary polynomial identity. In fact, I conjecture that one only needs relations involving the action of bounded length in order to conclude such a result.******The second theme of my Proposal addresses the verbal subspaces of an algebra. A "verbal" subspace is the subspace generated by all the values a polynomial in an algebra. Apart from a few very special polynomials, this is a completely new area of investigation in noncommutative PI-theory. One of the first problems under consideration asks: if the verbal subspace is finite-dimensional, does it follow that the verbal subalgebra and verbal ideal generated by the verbal subspace is also finite-dimensional? There is an interesting dual to every verbal subspace: the "marginal" subspace generated by the "zeros" of a polynomial. *****