Mathematical Methods in Physics: Geometry and Group Theory 2

The course aims to provide the student with the bases for the study of
physical systems with continuous symmetries, through the systematic study
of Lie groups and their representations. The course provides important
knowledge to face the courses of Theoretical Physics, Theory of
Fundamental Interactions and Gravity and Superstrings.

At the end of the course the student:

1) will be able to handle some basic notions of differential geometry:
manifolds, tangent spaces and bundles, vector fields, differential forms
2) will know the notions of Lie group and Lie algebra and the relationship
between them. He will also know the notions of one parameter subgroup,
exponential map, adjoint representation, Killing form
3) will know the classification of complex semisimple Lie algebras and the
notions of Cartan subalgebra, root, Dynkin diagram, real form of a complex
algebra
4) will know and know how to handle the representation theory of
semisimple Lie algebras and the weight diagrams. He will know the
relationship between the representations of a Lie algebra and those of the
associated Lie group
5) will be able to handle the products of representations
6) will know how to decompose the representations of algebras in terms of
representations of subalgebras
7) will know in detail some groups with particular relevance in physics:
the unitary groups U(N), the orthogonal groups O(N), the Lorentz group and
the Poincaré group, whose representations are classified by mass and spin.