LIE GROUPS AND LIE ALGEBRAS

The following sections are included:

• Lie Algebras and its Structure Constants

  • The Global Property of a Lie Group

  • The Local Property of a Lie Group

  • The Lie Algebra

  • The Killing Form and the Cartan Criteria

• The Regular Form of a Semisimple Lie Algebra

  • The Inner Product in a Semisimple Lie Algebra

  • The Cartan Subalgebra

  • Regular Commutative Relations of Generators

  • The Inner Product of Roots

  • Positive Roots and Simple Roots

• Classification of Simple Lie Algebras

  • Angle between Two Simple Roots

  • Dynkin Diagrams

  • The Cartan Matrix

• Classical Simple Lie Algebras

  • The SU(N) Group and its Lie Algebra

  • The SO(N) Group and its Lie Algebra

  • The USp(2ℓ) Group and its Lie Algebra

• Representations of a Simple Lie Algebra

  • Representations and Weights

  • Weight Chain and Weyl Reflections

  • Mathematical Property of Representations

  • Fundamental Dominant Weights

  • The Casimir Operator of Order 2

• Main Data of Simple Lie Algebras

  • Lie Algebra A_ℓ and Lie Group SU(ℓ + 1)

  • Lie Algebra B_ℓ and Lie Group SO(2ℓ + 1)

  • Lie Algebra C_ℓ and Lie Group USp(2ℓ)

  • Lie Algebra D_ℓ and Lie Group SO(2ℓ)

  • Lie Algebra G₂

  • Lie Algebra F₄

  • Lie Algebra E₆

  • Lie Algebra E₇

  • Lie Algebra E₈

• Block Weight Diagrams

  • Chevalley Bases

  • Orthonormal Basis States

  • Method of Block Weight Diagram

  • Some Representations of A₂

  • Some Representations of C₃

  • Planar Weight Diagrams

• Clebsch–Gordan Coefficients

  • Representations in the CG Series

  • Method of Dominant Weight Diagram

  • Reductions of Direct Product Representations in A₂

• Exercises