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Lie groups, Lie algebras and their representations (MATHM0012)

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Administrative Information

  1. Unit number and title: MATHM0012 Lie groups, Lie algebras and their representations
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2012
  6. Unit Organiser: Jonathan Robbins
  7. Lecturer: Jonathan Robbins
  8. Teaching block: 1
  9. Prerequisites: MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH11007 (Calculus 1), MATH20900 (Calculus 2). Students will be expected to have attained a degree of mathematical maturity and facility at least to the standard of a beginning level 7 student.

Unit aims

The aims of this unit are to introduce the principal elements of semisimple Lie groups, Lie algebras and their representations, for which there is a relatively complete and self-contained theory. The course will develop conceptual understanding as well as facility with calculation. By treating semisimple Lie groups as sets of finite-dimensional matrices (the alternative, more abstract point of view is to treat them as differentiable manifolds), the unit will be made accessible to a students with a broad range of backgrounds.

General Description of the Unit

Lie groups and Lie algebras embody the mathematical theory of symmetry (specifically, continuous symmetry). A central discipline in its own right, the subject also cuts across many areas of mathematics and its applications, including geometry, partial differential equations, topology and quantum physics. This unit will concentrate on finite-dimensional semisimple Lie groups and Lie algebras and their representations, for which there exists a rather complete and self-contained theory. Applications will be discussed. Students  will be expected to have attained a degree of mathematical maturity and facility at least to the standard of a beginning level 7 student.

NOTE: This unit is also part of the Oxford-led Taught Course Centre (TCC), and is taken by first- and second-year PhD students in Bristol and its TCC partner departments.  The unit has been designed primarily with a postgraduate audience in mind.  Undergraduate students should not normally take more than one TCC unit per semester. 

Relation to Other Units

This unit complements the following units:

 

  • Group Theory (MATH33300) and Representation Theory (MATH M4600) deal mainly with finite groups, but there are some strong parallels with the theory of compact Lie groups.
  • Differentiable Manifolds 3 (MATH 32900) and Differentiable Manifolds 34 (MATH 2900) provide the framework for the geometry of Lie groups. In turn, Lie groups, and spaces on which they act, constitute an important class of differentiable manifolds
  • Mechanics 2, Mechanics 23 (MATH 31910), Quantum Mechanics (MATH 35500), Quantum Chaos (MATH 5700), Quantum Information Theory (MATH 5600).  As the basis for the mathematical theory of continuous symmetries, Lie groups and Lie algebras play a prominent role in physics, particularly in classical mechanics and especially in quantum mechanics. 
  • Algebraic Topology (MATH 1200). Many Lie groups have interesting topological properties, and a number of central results in algebraic topology concern Lie groups.
  • Random Matrix Theory (MATH 33720) concerns the statistical distribution of eigenvalues of sets of matrices averaged over the action of certain Lie groups with respect to Haar measure.
  • Lie groups also appear in the syllabus of the new unit, 'Topics in Modern Geometry'. The point of view as well as the specific content in that unit will be independent of and complementary to the material covered in this one.
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Teaching Methods

The unit will be delivered through lectures.  Weekly problem sheets will be assigned and marked, and solutions distributed.

Learning Objectives

A student successfully completing this unit will be able to:

  • state the definition of matrix Lie groups and Lie algebras, explain the connections between them, and describe their relationship to symmetry;
  • state the definition of semisimple Lie groups and Lie algebras;
  • delineate the principal examples; formulate and apply the Cartan criterion for determining semisimplicity; construct examples of non-semisimple Lie algebras;
  • formulate the definitions of representation, irreducible representation and complete reducibility;
  • prove and apply Schur’s Lemma, formulate and apply a procedure for reducing a given representation into irreducible components;
  • explain the principal elements of the representation theory of finite-dimensional semisimple Lie algebras, including the Killing form, adjoint representation, Cartan subalgebra, and weights and roots;
  • explain and construct Dynkin diagrams;
  • give a complete classification of finite-dimensional semisimple Lie algebras, including theexceptional Lie algebras, and prove the principal theorems required for the classification;
  • define Haar measure; calculate Haar measure and evaluate integrals over groups in specific examples;
  • explain and apply the principal elements of the representation theory of compact semisimple Lie groups, including unitarity, orthogonality and completeness, and prove the principal theorems;
  • define tensor product representations, and decompose tensor products into irreducible components;
  • explain and apply some aspects of the representation theory of noncompact semisemple Lie groups in specific examples;
  • appreciate how the subject relates to some other areas of mathematics and physics, including, for example, differential geometry, partial differential equations, and/or quantum mechanics and quantum information theory;
  • apply results from the unit to problems in these areas.

Assessment Methods

The final assessment mark will be based on a 1½-hour written examination (100%).

Award of Credit Points

Credit points for the unit are gained by passing the unit (i.e. getting a final assessment mark of 50 or over).

Texts

H Georgi, Lie Algebras and Particle Physics, 2nd edition (1999)
B Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer-Verlag (2004)
JJ Duistermaat and JAC Kolk, Lie groups, Springer-Verlag (2000)

Syllabus

  1. Examples of matrix Lie groups and Lie algebras.  Representations.  Irreducibility.
  2. Semisimple Lie algebras. Killing form, Cartan subalgebra, roots and weights.  Dynkin diagrams.  Classification.
  3. Compact Lie Groups.  Haar measure.   Representations: complete reducibility, orthogonality, Peter-Weyl theorem.  Tensor product representations.
  4. Examples of non-compact matrix Lie groups and their representations - SL(2,R)