Groups, rings, lie and hopf algebras

我们继续使用各种技巧来研究代数的分次：结合超代数、李超代数、Jordan超代数和各自的超代数。这些技术大多与Hopf代数、超代数和群方案相联系，并且在任意特征域上的有限维代数的情况下有效地工作。我们将继续使用函数恒等式，这是一个有效的工具，在这种情况下，足够大的维代数，并且系数不需要在一个域中。我们将更多地关注各种类型的非结合代数，这将有助于我们研究例外李代数的分次。我们计划继续研究结合代数和李代数上的分次恒等式，因为它们被证明是普通多项式恒等式的有效集合，而普通多项式恒等式即使在最简单的矩阵代数中也很难确定。我们将继续研究各种代数对象的增长，如群，结合代数和李代数，它们的作用和模，目的是构造通过它们的增长接近自由的对象，并且仍然满足有趣的有限条件。这对于Burnside类问题的研究中出现的构造是重要的。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，记住无限维单李代数，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.