Lie Algebras and Lie Groups

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

The aim of the lectures is get students familiar with the basic concepts of the theory of Lie groups and Lie algebras, and their finite-dimensional representations. The students will also learn Cartan's classification of simple complex Lie algebras, which is the fundamental result in this field of mathematics, including its derivation. Emphasis is put on detailed investigation of explicit examples of the introduced mathematical structures and their applications.

Requirements:

Syllabus of lectures:

1. Overview of basic notions of differential geometry: differentiable manifold, tangent vectors, tangent space, vector field and its integral curves, commutator of vector fields, tangent map, exterior differential algebra, exterior and Lie derivative, Poincaré lemma, pullback of forms.

2. Lie group and Lie algebra - definitions, exponential mapping, flow of levoinvariant vector field.

3. Matrix groups and algebras, ambiguity in relation between Lie groups and algebras, classification of

connected Lie groups with a given Lie algebra.

4. Subgroups and subalgebras, actions of groups, cosets, isotropy subgroup, homogeneous spaces, examples of

spaces and spacetimes with transitive actions as homogeneous spaces.

5. Representation of Lie group / algebra, adjoint representation, irreducibility of representations, Schur lemma,

examples of completely reducible representations.

6. Basic classes of Lie algebras, Levi decomposition theorem into radical and semisimple Levi factor,

classification of Lie algebras over R and C in dimensions 1, 2, 3 and their properties

7. Nilpotent and solvable Lie algebras, Engel theorem and its formulation for matrix Lie algebras, Lie theorem,

properties of derivable algebra

8. Killing's form, Cartan criteria of semisimplicity and solvability of the given algebra, decomposition of

semisimple algebras into simple ideals

9. Cartan subalgebra and root system, their properties, Weyl-Chevalley normal form of semisimple Lie algebra,

classification of simple Lie algebras over C, root and Dynkin diagrams

10. Finite-dimensional representations of simple Lie algebras over C, weights and weight diagrams, group SU(3)

and its applications for elementary particle classification.

Syllabus of tutorials:

Examples of mathematical structures defined during lectures, their use in mathematics and theoretical physics, detailed evidence of some simpler statements from lectures.

Study Objective:

Study materials:

Key references:

[1] R. Gilmore: Lie Groups, Physics and Geometry, Cambridge University Press 2008.

[2] A. P. Isaev, V. A. Rubakov: Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras, World Scientific 2018.

Recommended references:

[3] L. Šnobl, P. Winternitz: Classification and Identification of Lie Algebras, American Mathematical Society 2014.

[4] D. H. Sattinger, O.L. Weaver: Lie Groups and Algebras, Springer Verlag 1986.

[5] K. Erdmann, M.J. Wildon: Introduction to Lie Algebras, Springer Verlag 2006.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Matematická fyzika (compulsory course in the program)