Cartan criteria for solvability and semisimplicity. Semisimple Lie algebras ( = radical is 0). Nilpotent Lie algebras and nil-representations. Universal enveloping algebras and the PBW theorem (this may be postponed or/and dissolved in the future lectures.).From Lie G to G: Campbell-Hausdorff, local Lie groups, Ado theorem (without a proof, at least here).Here (or before) a theorem is proved: every closed subgroup of a Lie group is a Lie subgroup. Relation between Lie subgroups of G and Lie subalgebras of Lie G.(The goal here is to reduce the theory of Lie groups to the theory of Lie algebras the latter will be handled in Part II.) Representation of Lie groups and Lie algebras. Homomorphisms ϕ: G1 → G2 and dϕ: Lie G1 → Lie G2 – all relations between them.One-parameter subgroups, exponential map TeG → G (the case of matrices).Products, coverings, Lie subgroups, quotients (homogeneous spaces).Lie groups, Lie algebras, and basic relations between them.
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