Università di Pavia - Offerta formativa

LIE GROUPS AND REPRESENTATION THEORY

Enrollment year

2020/2021

Academic year

2020/2021

Regulations

DM270

Academic discipline

FIS/02 (THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS)

Department

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

Course

MATHEMATICS

Curriculum

PERCORSO COMUNE

Year of study

1°

Period

2nd semester (01/03/2021 - 11/06/2021)

ECTS

6

Lesson hours

48 lesson hours

Language

Italian

Activity type

ORAL TEST

Teacher

BISIO ALESSANDRO (titolare) - 6 ECTS

Prerequisites

The knowledge of the basic mathematical tools, typically learned during the BSc courses, is required. It is desirable that the student has studied or is studying differential geometry.

Learning outcomes

Learning the basic notions and concepts both of the theory of Lie groups, including homogeneous manifolds and of that of Lie algebras, including their classification. Particular emphasis will be given to the theory of representations on Hilbert spaces.

Course contents

In the first part of the course, the basic structural properties of Lie groups are introduced, in particular the related notion of Lie algebra and its geometrical interpretation. Then, we introduce and study in detail the exponential map. In the second part of the course we prove the Frobenius theorem and we use it both to construct homogeneous manifolds starting from a Lie group and to study their geometric properties. Subsequently the theory of Lie algebras is presented and the Cartan classification of semisimple Lie algebras is outlined. During the third and last part of the course, we develop the theory of representations of Lie groups on Hilbert spaces, giving emphasis to the applications in quantum mechanics and in quantum field theory.

Teaching methods

On account of the theoretical and mathematical nature of the course, all classes will be given at the blackboard and all topics of the programme will be discussed.

Reccomended or required readings

F. Warner "Foundations of differentiable manifolds and Lie groups" (1990) 3ed. Springer-Verlag.

A. W. Knapp "Lie groups: Beyond an introduction" (2005) BirkhĂ¤user

A. O. Barut, R. Raczka "Theory of Group representations and applications" (1986) World Scientific.

A. W. Knapp "Lie groups: Beyond an introduction" (2005) BirkhĂ¤user

A. O. Barut, R. Raczka "Theory of Group representations and applications" (1986) World Scientific.

Assessment methods

The student has to give an oral exam aimed at verifying that he/she has learned the topics presented during the lectures. Special emphasis will be given to the verification of the ability of the student to present rigorously and with an appopriate technical language all concepts learned.

Further information

The student has to give an oral exam aimed at verifying that he/she has learned the topics presented during the lectures. Special emphasis will be given to the verification of the ability of the student to present rigorously and with an appopriate technical language all concepts learned.

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