The structure theory of Lie groups and Lie algebras: left-invariant vector fields, the classical groups as manifolds, the exponential map and where it fails to be surjective, the Baker-Campbell-Hausdorff formula, the Lie correspondence and covering groups, the Killing form and Cartan's criteria, Engel's and Lie's theorems, the Levi decomposition, the universal enveloping algebra and PBW, Casimir elements, maximal tori, the Weyl character and dimension formulas, real forms and the unitary trick, homogeneous and symmetric spaces, and symmetry in physics. Representation theory itself is developed in the Representation Theory chapter.
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