Lie and Jordan algebras, their representations and generalizations

Abstract

The main part of the project is devoted to Lie and Jordan algebras and superalgebras, which are the principal classes of non associative algebras. Besides, alternative and Malcev algebras and superalgebras will be considered, likewise Moufang loops and various generalizations of algebras mentioned above. The main directions of research will be the following: 1) structure and representations of algebras; 2) geometrical aspects of the theory of algebras; 3) combinatorial aspects of the theory of algebras; 4) applications and generalizations. The typical questions in the first direction are related with the study and classification of simple structures (basic atomic building blocks out of which all other structures are built): simple and prime algebras and superalgebras, irreducible and indecomposable modules, Verma modules, etc. Besides, the specialty problem will be studied for Jordan and Malcev algebras. In the second direction, the structure of certain algebraic varieties related with finite dimensional algebras and with various categories of representations will be investigated. The combinatorial part of the theory of algebras has many common features in associative and non-associative cases, both in the problems and in the methods of their solution. For example, the theory of PI-algebras played an important role in the classification of simple alternative and Jordan algebras. On the other hand, the algebras of Poisson brackets (which are non-associative) proved to be very useful in the study of automorphisms of polynomial algebras. Therefore, in this part of the research we consider both associative and non-associative algebras. We plan to study automorphisms and subalgebras of free algebras, polynomial identities of algebras and superalgebras from various classes. Besides, we will try to extend to non-associative algebras some results from the theory of Hopf algebras. Finally, in the part of applications and generalizations, we will study Moufang loops, which have strong relations with alternative algebras, and some generalizations of Jordan algebras. (AU)