The principal investigator intends to apply model-theoretic methods to the representation theory of Lie algebras and noncommutative rings, as well as to the K-theory of a pointed abelian category. One of the main goals of the project will be to generalize classical results about finite-dimensional representations to the pseudo-finite dimensional case - a representation of an algebra is called pseudo-finite dimensional if it satisfies the axioms for a finite-dimensional representation. A theory of pseudo-finite dimensional representations has already been introduced for the Lie algebra sl (2,k), and the project is devoted to generalizing this theory for the case of a finite-dimensional semisimple Lie algebra L , as well as the corresponding Lie group G(L). The project will also focus on the complex of positive-primitive formulae in the language of modules over an associative ring R. The aim will be to establish a relationship between the homology of this complex and the K-theory of the free abelian category over R.
Symmetry plays an essential role in the study of physical objects that occur in Nature. The symmetries of an object - think, for example, of a crystal or a butterfly - are the rigid motions that bring the object back onto itself. These symmetries form an algebraic structure, called a group. In this way, we can associate to every physical object its group of symmetries. In representation theory, the mathematician considers a group G as an abstract object and attempts to classify all the objects of which G is the symmetry group. Traditionally, research has focused on the case where the objects are finite-dimensional vector spaces. In this project, the methods of mathematical logic are applied to include the infinite-dimensional cases called pseudo-finite dimensional. These are the situations that the limited power of expression of the mathematical language in use cannot distinguish from the finite-dimensional. It has been shown that many such objects exist, but to unlock the secrets behind any particular one of them remains a baffling question.
