多项式代数及自由结合代数的自同构和导子

本项目围绕多项式自同构的tame生成子问题，研究多项式代数和自由结合代数的自同构和导子。（1）描述能和仿射群共同生成多项式自同构群或tame自同构群的自同构集合；利用tame自同构的多重次数集合的结构研究II，III型约化的存在性问题。（2）利用PI-环理论研究多项式自同构到自由结合代数的提升问题以及可提升性与tame性的关系。（3）研究多项式导子的Darboux多项式、常数环、三角化以及特殊局部幂零导子及其诱导的指数自同构，构造有用的例子和反例。

Around Tame Generators Problem this project investigates automorphisms and derivations of polynomial algebras and free associative algebras. (1) We intend to describe sets S of automorphisms such that S with the set of affine automorphisms generates the group of polynomial automophisms or the group of tame automorphisms. Using the structure of sets of multidegrees for tame automophisms studies the problem on existence of reduction of type II and III. (2) By means of theory of PI-rings we investigate the problem on lifting polynomial automophisms to ones of free associative algebras and how it is related to tameness. (3) We investigate Darboux polynomial, ring of constants, triangularization for polynomial derivations, locally nilpotent derivations and the automophisms induced by them to provide useful examples and counterexamples.