Mathematics 253/Physics 293: Representation Theory (Fall 2003)

Instructor

Description

• Finite Groups
  • Basic definitions
  • Schur's Lemma
  • Characters
  • Induced representations
  • Real representations
  • The symmetric groups and Young Diagrams
• Lie Groups and Lie Algebras
  • Basic definitions
  • Basic notions of classification
  • Simple and semisimple Lie algebras
  • Trivial low dimensional examples
  • sl(2) in gory detail
  • sl(3) in gory detail
• Classical Lie Algebras
  • General constructions and the Killing form
  • sl(n) and Young diagrams again
  • sp(n)
  • so(n) and spinors
• The General Classification
  • Dynkin diagrams
  • The exceptional algebras g2, f4, e6, e7, e8
  • Characters
  • Lie groups

Representation theory studies how groups or algebras can act as linear transformations on vector spaces. This course is concerned mainly with finite groups and semisimple Lie algebras over the complex numbers. The material in this course is important for students interested in Algebra, Algebraic Geometry, Differential Geometry, Mathematical Physics, Number Theory, and Topology. This course is cross-listed with the physics department as Physics 293 and is often taken by physics graduate students. In addition this course is a prerequisite for the course on Lie Groups and Symmetric Spaces (often taught as Math 268).

Prerequisites

Text(s)

• W. Fulton and J. Harris, Representation Theory: A First Course, Springer-Verlag 1991.

Course Website

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