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Module MA3415: Introduction to Lie Algebras
- Credit weighting (ECTS)
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5 credits
- Semester/term taught
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Michaelmas term 2015-16
- Contact Hours
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11 weeks, 3 lectures including tutorials per week
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- Lecturer
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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
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- 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
- MA2215. Desirable MA2322 & MA3429
Literature:
- Humphreys, Introduction to Lie algebras and representation theory.
- Serre, Lie algebras and Lie groups.
- Carter, Lie algebras of finite and affine type.
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- Assessment Detail
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This module will be examined in a 2 hour examination in Trinity term.
Continuous assessment
will contribute 10% to the final grade for the module at the annual
examination session. Supplementals if required will be assessed 100% exam.