Mathematics 253/Physics 293: Representation Theory (Fall 2005)

Instructor

Description

• Representations of Finite Groups
  • Definitions
  • Schur's Lemma
  • Characters
  • Induced representations
  • Real representations
  • Symmetric groups and Young Diagrams
• Lie Groups and Lie Algebras
  • Definitions and examples
  • Lie theory and the subgroup/subalgebra correspondence
  • Solvable and nilpotent Lie algebras
  • Simple and semisimple Lie algebras
  • Representations of sl₂C
  • Representations of sl₃C
• Structure and Representations of Arbitrary Semisimple Lie Algebras
  • The Killing form
  • sl_nC
  • sp_nC
  • so_nC and spinors
• Classification of Complex Semisimple Lie Algebras
  • Dynkin diagrams
  • Exceptional algebras g₂, f₄, e₆, e₇, e₈
• The Weyl Character Formula

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.

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