# C2.1 Lie Algebras - Material for the year 2020-2021

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B2.1 Introduction to Representation Theory is recommended. A thorough knowledge of linear algebra and familiarity with group actions, quotient rings and vector spaces, isomorphism theorems and inner product spaces will be assumed. Some familiarity with the Jordan-HÃ¶lder theorem and the general ideas of representation theory will be an advantage.

16 lectures.

### Assessment type:

- Written Examination

Lie Algebras are mathematical objects which, besides being of interest in their own right, elucidate problems in several areas in mathematics. The classification of the finite-dimensional complex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are to introduce Lie algebras, develop some of the techniques for studying them, and describe parts of the classification mentioned above, especially the parts concerning root systems and Dynkin diagrams.

By the end of the course, students will be able to

- Identify the basic classes of Lie algebras - nilpotent, solvable and simisimple, and be able to give examples of them.
- Use basic notions such as ideals and representations to analyse the structure of Lie algebras, and employ the Cartan criteria in problems.
- Use the theory developed in the course to calculate the Dynkin Diagram of (semi)simple Lie algebra in examples.

Definition of Lie algebras, small-dimensional examples, some classical groups and their Lie algebras (treated informally). Ideals, subalgebras, homomorphisms, modules.

Nilpotent algebras, Engel's theorem; soluble algebras, Lie's theorem. Semisimple algebras and Killing form, Cartan's criteria for solubility and semisimplicity, Weyl's theorem on complete reducibility of representations of semisimple Lie algebras.

The root space decomposition of a Lie algebra; root systems, Cartan matrices and Dynkin diagrams. Discussion of classification of irreducible root systems and semisimple Lie algebras.

- J. E. Humphreys,
*Introduction to Lie Algebras and Representation Theory,*Graduate Texts in Mathematics 9 (Springer-Verlag, 1972, reprinted 1997). Chapters 1-3 are relevant and part of the course will follow Chapter 3 closely. - B. Hall,
*Lie Groups, Lie Algebras, and Representations. An Elementary Introduction,*Graduate Texts in Mathematics 222 (Springer-Verlag, 2003). - K. Erdmann, M. J. Wildon,
*Introduction to Lie Algebras*(Springer-Verlag, 2006), ISBN: 1846280400.

- J.-P. Serre,
*Complex Semisimple Lie Algebras*(Springer, 1987). Rather condensed, assumes the basic results. Very elegant proofs. - N. Bourbaki,
*Lie Algebras and Lie Groups*(Masson, 1982). Chapters 1 and 4-6 are relevant; this text fills in some of the gaps in Serre's text. - William Fulton, Joe Harris,
*Representation theory: a first course*, GTM, Springer.