Applications of Model Theory to Representation Theory

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The principal investigator intends to apply model-theoreticmethods to the representation theory of Lie algebras andnoncommutative rings, as well as to the K-theory of a pointedabelian category. One of the main goals of the project will beto generalize classical results about finite-dimensionalrepresentations to the pseudo-finite dimensional case - arepresentation of an algebra is called pseudo-finite dimensionalif it satisfies the axioms for a finite-dimensionalrepresentation. A theory of pseudo-finite dimensionalrepresentations has already been introduced for the Lie algebrasl (2,k), and the project is devoted to generalizing this theoryfor the case of a finite-dimensional semisimple Lie algebra L ,as well as the corresponding Lie group G(L). The project willalso focus on the complex of positive-primitive formulae in thelanguage of modules over an associative ring R. The aim will beto establish a relationship between the homology of this complexand the K-theory of the free abelian category over R.Symmetry plays an essential role in the study of physical objectsthat occur in Nature. The symmetries of an object - think, forexample, of a crystal or a butterfly - are the rigid motions thatbring the object back onto itself. These symmetries form analgebraic structure, called a group. In this way, we canassociate to every physical object its group of symmetries. Inrepresentation theory, the mathematician considers a group G asan abstract object and attempts to classify all the objects ofwhich G is the symmetry group. Traditionally, research hasfocused on the case where the objects are finite-dimensionalvector spaces. In this project, the methods of mathematical logicare applied to include the infinite-dimensional cases calledpseudo-finite dimensional. These are the situations that thelimited power of expression of the mathematical language in usecannot distinguish from the finite-dimensional. It has been shownthat many such objects exist, but to unlock the secrets behindany particular one of them remains a baffling question.