## Trinity College Dublin, The University of Dublin

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# Module MAU34103: Introduction to Lie Algebras

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2019-20
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 repesentation of sl2.
• State the fundamental theorem of Lie theory, PBW theorem, Engel's theorem, Lie's theorem and Cartan's criterion.
• Describe the Jordan-Chevalley 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 Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. Re-assessments if required will consist of 100% exam.