Descripción
**Fecha de inicio**: 19/07/2010. **Fecha de fin**: 21/07/2010.
Abstract : The aim of my lectures is to give an introduction to one-parameter
formal deformation theory of algebras and to give an overview of the
main ideas and results in deformation quantization. The Deformation
theory has been applied as a useful tool of many algebraic structures. The fundamental results of
Gerstenhabers theory connect deformation theory with the suitable
cohomology groups. We will provide in this course some basics of homological algebra and describe in particular the Gerstenhaber algebra
of the cochains and introduce the Schouten bracket.
Since the seminal article by Bayen, Flato, Fronsdal, Lichnerowicz et
Sternheimer in 1978, deformation quantization has become a large
research area
covering several algebraic theories like the formal deformation theory
of associative algebras and the more recent theory of operads, as well
as geometric
theories like the theory of symplectic and (more generally) Poisson
manifolds, and of physical theories like string theory and
noncommutative gauge
theory. In this theory, the noncommutative associative multiplication of
operators in quantum mechanics is considered as a formal associative
deformation
of the pointwise multiplication of the algebra of symbols of these
operators.
The course will be self contained and avoid a too technical
considerations. Many examples will be supplied.