**Lie theory**

UPDATED Lie theory course available as a ★★★★★ PDF text. Archived Lie theory course:

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

**Diffeomorphisms and smooth manifolds**

**Miscellaneous geometries**

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