Groups, rings, lie and hopf algebras

我们继续使用各种技术来研究代数上的分级：结合代数、李代数、乔丹代数和各自的超代数。这些技术大多与Hopf代数、超代数和群方案有关，并且在任意特征域上的有限维代数的情况下有效地工作。我们将继续使用泛函恒等式，这是一个有效的工具，在这种情况下，代数的维数足够大，系数不需要在一个域中。我们将更多地关注各类非结合代数，这将有助于我们研究例外李代数的评分。我们计划继续研究结合代数和李代数上的分级恒等式，因为它们被证明是普通多项式恒等式的有效替代，即使在最简单的矩阵代数情况下也很难确定。我们将继续研究各种代数对象的增长，如群、结合子代数和李代数，它们的作用和模块，目标是通过它们的增长构造接近自由的对象，并且仍然满足有趣的有限条件。这对于研究burnside型问题时产生的结构是很重要的。我们将继续研究无限维李代数和超代数，特别是局部有限代数，考虑无限维简单李代数，它们的结构和表示。

We continue to use various techniques to study gradings on algebras: associative, Lie, Jordan and respective superalgebras. Most of these techniques are connected to Hopf algebras, hyperalgebras and group schemes, and work effectively in the case of finite-dimensional algebras over fields of arbitrary characteristic. We will continue using functional identities, which are an effective tool in the case, where the algebras of sufficiently large dimension, and the coefficients need not be in a field. We will pay more attention to various classes of nonassociative algebras which will help us in the study of gradings on exceptional Lie algebras. We plan to continue studying graded identities on associative and Lie algebras since they turn out to be an effective sustitute of ordinary polynomial identities which are very hard to determine even in the simplest cases of matrix algebras. We will continue our research of the growth of various algebraic objects, like groups, associative and Lie algebras, their actions and modules, with the goal of constructing the objects close to free by their growth and still satisfying interesting finiteness conditions. This is important for the constructions arising in the study of Burnside-type problems. We will continue working in the area of infinite-dimensional Lie algebras and superalgebras, in particular, locally finite algebras, having in mind infinite-dimensional simple Lie algebras, their structure and representations.