Lie theory

  1. Introduction to Lie groups
  2. Lie Bracket, closure under the Lie Bracket
  3. Derivations and the Jacobi identity
  4. Lie group homomorphisms
  5. Exponential map: injectivity and surjectivity, diffeo, examples of differential correspondence (exp, Taylor, derivations, adjoint), continuous maps are smooth [1], Baker-Campbell-Hausdorff
  6. Lie group topology
  7. Lie correspondence (exercise: determinant unique)
  8. The Killing form; factorising non-Abelian Lie groups
  9. Some applications (parallel parking, indefinite Orthogonal group, some isomorphisms, some connected and compact proofs, odd-dimensional spheres, covers and quaternions)
  10. Abstract Lie algebra, representation theory [3]
  11. Lie derivatives, relations to differential geometry [4]
  12. Lie Geometry -- [5]; exponential surjective when compact

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