Groups, rings, lie and hopf algebras

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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.