Graded algebras and applications

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Group gradings of algebras play an important role in mathematics. Classical example are Cartan decomposition of classical Lie algebras, and Cayley-Dickson gradings of composition algebras. Both are widely used in Lie Theory, a tool of many working in Pure and Applied Mathematics and in Mathematical Physics. Gradings also provide deformations of algebras, important in searches of objects describing physical models.******Our work on the classification of gradings on simple algebras started more than two decades ago, and now involves specialists in at least 10 countries of the world. In the case of algebraically closed fields, the classification is essentially complete in the case of finite-dimensional algebras. The methods include classical algebraic groups but also approaches suggested my colleagues and myself, involving Hopf algebras and algebraic group schemes. The technique of so called functional identities, developed jointly with M. Bresar, allowed us to handle infinite-dimensional algebras. ******In the case of arbitrary fields, one needs to know graded-division algebras, since graded-simple algebras can be represented by linear operators of graded vector spaces over graded-division algebras. In the most "practical" case of real numbers, our recent papers with M. Kochetov, A. Rodrigo-Escudero and M. Zaicev provide a good basis for the theory of real graded-simple algebras, with possible applications to Differential Geometry. We have already started work on the gradings of algebras over arbitrary fields, and fields themselves; rich theory of fields and division algebras, will enable us to enrich the theory with new tools and interesting examples. ******An important application is the graded modules over Lie algebras. Recently, A. Elduque and M. Kotchetov have published results on the possibility of grading finite-dimensional irreducible modules of classical simple Lie algebras. Now, with M. Kochetov and A. Shihadeh, we work on providing explicit grading to those representations and on infinite-dimensional representations. Providing gradings will clarify the structure of these modules and will be useful for those working in this popular area. ******One more directions is the theory of PI-algebras. For most simple algebras, their ordinary identities are not known. Graded identities are much easier, yet they define ordinary identities. In a work with F. Yasumura, we proved that graded-simple finite-dimensional algebras over an algebraically closed field, with the same graded identities, are isomorphic as graded algebras. We will be working on the extension of this to other situations and to the embeddings of algebras, in place of isomorphisms. ******In our work with Susan Montgomery, we use our extensive knowledge of the gradings on simple algebras to the study of actions of Hopf algebras, with sufficiently large groups of group-like elements. Starting with Taft algebras and their Drinfeld doubles, we will explore new situations where this approach works.**