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.
Expected learning outcomes
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.
In the course, Lie Groups and Lie Algebras are studied and various aspects of the theory of their representations are discussed.
- Differential geometry: manifolds, tangent spaces and bundles, vector fields, differential forms - Lie groups and Lie algebras. Relationship between algebra and group. One parameter subgroups. Exponential map. Adjoint representation. Killing form. - Classification of complex semisimple Lie algebras: Cartan subalgebras, roots, Dynkin diagrams. Real forms. - Irreducible representations of semisimple Lie algebras. Relations between representations of a Lie algebra and those of the associated Lie group. Weights. Products of representations. - Subalgebras and subgroups. Branching rules of representations. - Analysis of symmetries with particular relevance in physics. The unitary groups U(N) and orthogonal groups O(N). The Lorentz group. The Poincaré group: classification of its representations by means of mass and spin.
Prerequisites for admission
Knowledge of mathematical analysis and linear algebra.
Lectures on the blackboard with theoretical discussion and exercises on the covered topics.
MATERIALE DI RIFERIMENTO:
- F. Warner, "Foundations of Differentiable Manifolds and Lie Groups" - J.F. Cornwell, "Group Theory in Physics", Vol.1 and Vol.2 - A. Arvanitoyeorgos, "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces"
Assessment methods and Criteria
Exercises to be carried out at home and oral exam with discussion of the solutions and of the topics covered in class.