Mathematical methods in physics: geometry and group theory 2

A.Y. 2020/2021
Overall hours
Learning objectives
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
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.
Course syllabus and organization

Single session

Lesson period
First semester
The course will be delivered entirely remotely in case of travel
restrictions due to Covid-19. In this case, the lectures will be offered
in virtual classrooms (zoom platform) in synchronous connection, with the
possibility of real-time interaction between the students and the teacher.
Course syllabus
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
- 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.
Teaching methods
Lectures on the blackboard with theoretical discussion and exercises on
the covered topics.
Teaching Resources

- 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.
Lessons: 42 hours