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Module MAU34103: Introduction to Lie Algebras
 Credit weighting (ECTS)

5 credits
 Semester/term taught

Michaelmas term 202122
 Contact Hours

11 weeks, 3 lectures including tutorials per week

 Lecturer

Prof. Sergey Mozgovoy
 Learning Outcomes
 On successful completion of this module, students will be able to
 Give the definitions of: Lie group, Lie algebra, exponential map, homomorphism of Lie algebras, representation of a Lie algebra, subrepresentation, irreducible representation, homomorphism of representations, universal enveloping algebra, the Killing form of a Lie algebra, nilpotent Lie algebra, solvable Lie algebra, semisimple and simple Lie algebra, Cartan subalgebra, root system.
 Give the definitions of and calculate with the classical Lie algebras.
 Describe the construction of the irreducible representation of sl2.
 State the fundamental theorem of Lie theory, PBW theorem, Engel's theorem, Lie's theorem and Cartan's criterion.
 Describe the JordanChevalley decomposition for semisimple Lie algebras.
 Give the root space decomposition and root system of sln.
 Module Content

 Lie groups, Lie algebras, examples.
 Lie algebra of a Lie group, exponential map, adjoint representation.
 Universal enveloping algebra, PBW theorem, Casimir element.
 Irreducible representation of sl2 (c).
 Nilpotent Lie algebras, engel's theorem.
 Semisimple Lie algebras, Killing form, Cartan's criterion.
 Complete reducibility (Weyl's theorem).
 Cartan decomposition of a semisimple Lie algebra.
 Irreducible representations of a semisimple Lie algebra.
 Module Prerequisite
 MAU11100, MAU23206
Literature:
 Humphreys, Introduction to Lie algebras and representation theory.
 Serre, Lie algebras and Lie groups.
 Carter, Lie algebras of finite and affine type.

 Assessment Detail

This module will be examined in a 2 hour examination in Michaelmas term.
Continuous assessment
will contribute 15% to the final grade for the module at the annual
examination session. Reassessments if required will consist of 100% exam.