| title: | Generalized exterior calculuses and N-complexes of Young symmetry type tensors |
|---|---|
| reg no: | ETF6206 |
| project type: | Estonian Science Foundation research grant |
| status: | accepted |
| institution: | University of Tartu |
| head of project: | Viktor Abramov |
| duration: | 01.01.2005 - 31.12.2006 |
| description: | There are the generalizations of the classical exterior calculus, which arised in the connection with the study of hypersymmetry. The classical exterior calculus of differential forms plays an essential role in the theory of smooth manifolds. An important part of the exterior calculus is exterior differential, which can be characterized by its nilpotency which means that the square of exterior differential equals zero. This nilpotency underlies the theory of de Rham cohomology on smooth manifold and Poincare duality. The main goal of the present project is to study the generalizations of a classical exterior calculus such that exterior differential of these generalizations satisfies the generalized nilpotency (N-niplotency), which means that if N is an arbitrary integer greater than 1 then exterior differential to power N equals zero. The second goal of the present project is to study the possible applications of the above mentioned generalizations of exterior calculus to a gauge field theory and to a theory of quantum spaces. There are two different generalizations of exterior calculus (both with N-nilpotent exterior differential) which we plan to study in the frame of the present project. The first one is based on the q-differential algebra notion. This approach has been elaborated and studied in the series of papers written by the author of this project. In this approach the classical finite dimensional space is replaced by an associative algebra whose generators satisfy certain commutation relations and one uses the corresponding generalized first order differential calculus. This formalism was recently used in order to construct the generalized exterior calculus with exterior differential cubed equals to zero on the two-parameter quantum plane and reduced quantum plane. The obtained results clearly show that the gauge field theory based on this generalized exterior calculus is very interesting and rich in content. We intend to study the two-parameter quantum plane and reduced quantum plane by means of more general exterior calculus, where the cubic nilpotency is replaced by more general N-nilpotency. The second generalization of exterior calculus, which we plan to study within this project, is based on Young diagrams and the Young symmetrizer. The main goals of the present project are the following: 1) to generalize recently constructed exterior calculus on the two-parameter quantum plane and the reduced quantum plane with exterior differential cubed equals to zero to exterior calculus with N-nilpotency by means of the apparatus developed by the author of this project, and to study the gauge field theories, which can be constructed on the base of this exterior calculus; 2) to study the generalization of exterior calculus based on the Young diagrams and corresponding symmetries on a smooth manifold constructing an exterior differential with the help of a sequence of connections; 3) to study the structure of an algebra of covariant tensors with certain Young type symmetries; 4) to apply a generalized exterior calculus based on the Young diagrams to the study of gauge field theories of higher spin and BRST-formalism. |
| project group | ||||
|---|---|---|---|---|
| no | name | institution | position | |
| 1. | Viktor Abramov | University of Tartu | assistant professor | |
| 2. | Piret Kuusk | Institute of Physics at University of Tartu | Senior research scientist, head of laboratory | |