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Important information

The time of the course is **Monday, 10:15, in lecture room H405A.**

We start on 4. February.

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Description and Syllabus

The aim of this course is to give an introduction into the theory of Lie groups, Lie algebras, and their representations, with an emphasis on applications in physics. The focus will be on concrete computations, therefore some of the mathematical theorems will not be rigorously proven, only motivated and illustrated. We present the classification of the simple complex Lie algebras in terms of root systems, Dynkin diagrams, and discuss the corresponding representation theory. We also discuss the real forms of the complex Lie algebras, leading to the classical Lie groups. A special focus will be given to the groups SU(2), SU(3), O(4), SO(3,1) (the Lorentz group), and the Poincaire group

Syllabus:

-Definitions: Lie groups, Lie algebras

-The Classical Lie groups (real and complex cases, including the orthogonal, unitary, and symplectic groups, and also the ,,indefinite'' cases such as the Lorentz group)

-The Poincaire group as a semi-direct product

-Representations: definitions, Schur's lemmas, unitary and non-unitary representations, Clebsch-Gordan series

-Representations of SU(2) and SO(3)

-Classification of simple complex Lie algebras (Cartan generators, Killing form, Dynkin diagrams)

-The real forms of the simple complex algebras, Cartan involutions

-Representations of simple complex Lie algebras: the algebraic approach, highest weight method

-Irreducible representations of SU(3) and SU(N) with the tensor method, Young diagrams

-Selected topics, depending on time and interest: Low dimensional coincidences, the n-dimensional harmonic oscillator, the symmetries of the Hydrogen atom, spinors and the Clifford algebra, symmetry breaking and branching rules

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Literature

-Christoph Lüdeling: Group Theory for Physicists. This is a very nice set of lecture notes. The present course is built in 60% on this.

-Peter Woit: Quantum Theory, Groups and Representations: An Introduction. A long and detailed book on the mathematical structures behind QM and QFT.

-Gerard t'Hooft: Lie groups in physics

-Howard Georgi: Lie Algebras in Particle Physics: from Isospin to Unified Theories