Algebras, representations e applications

Abstract

The main part of the project is dedicated to Lie and Jordan algebras and super algebras and their representations. Besides, alternative and Malcev algebras and super algebras will be considered, as well as Moufang loops and various generalizations of algebras mentioned above. The main directions of research will be the following: Lie algebras and their representations; Jordan and alternative algebras and super algebras and their generalizations; identities, graduations, automorphisms; applications e generalizations. Lie algebras and their representations is one of the most important and active areas of modern algebra, with an ample variety of results, applications and relations with other parts of the science. The new developments in non-commutative geometry and mathematical physics suggests a necessity to study infinite-dimensional modules, such as generalized Verma modules, modules with finite-dimensional weight spaces, Gelfand-Tsetlin and Harish¬Chandra modules, etc. The first part of the project treats various aspects of this theory. Besides the Lie algebras, the most important classes of non-associative algebras are alternative and Jordan algebras. The typical questions of the second part treat the study and classification of simple algebras and super algebras and irreducible and indecomposable modules in the varieties of alternative and Jordan algebras and their generalizations. The third part of the project is dedicated to combinatorial questions of the theory of algebras: study of free algebras, identities, gradations and automorphisms. This part of the theory has many common features in associative and non-associative cases, both in the problems and in the methods of their solution. Therefore, in this part of the research we consider both associative and non-associative algebras. We plan to study automorphisms and sub algebras of free algebras, identities and gradations of algebras and super algebras from various classes. Besides, we will try to extend to non-associative algebras some results from the theory of Hopf algebras. In the part of applications and generalizations, we will study Moufang loops, which have strong relations with alternative algebras and some other important classes of loops. We will also study the Albert Problem on the existence of simple commutative power associative finite dimensional nil algebras and will consider algebras that appear in genetics. (AU)