Уваров Ф.В. - Группы и Алгебры Ли - 15. Nilpotent Lie algebras, radical and semisimplicity

00:00:14 Plan of the lecture
00:00:26 Definition 1: a lower central series
00:02:26 Definition 2: nilpotent Lie algebra
00:03:34 Proposition 1: criterion of nilpotency
00:05:51 Proof of the proposition 1
00:07:17 Proposition 2: Any nilpotent Lie algebra is solvable
00:07:56 Proof of the proposition 2
00:11:23 Example of a nilpotent Lie algebra (strictly upper triangular matrices)
00:13:43 Proposition 3: "n" of a flag is nilpotent
00:15:11 Proof of the proposition 3
00:19:03 Proposition 4: J is solvable iff [J,J] is nilpotent
00:19:39 Proof of the proposition 4
00:32:14 Questions about proposition 4
00:33:04 Theorem 1: a basis of fin-dim vector space is strictly-upper triangular matrices
00:38:14 Engel's theorem
00:39:47 Proof of Engel's theorem
00:46:29 Definition 3: a semisimple Lie algebra
00:47:07 Definition 4: a simple Lie algebra
00:48:12 Proposition 5: if J is simple then it's semisimple
00:49:32 Proof of the proposition 5
00:51:49 Examples of semisimple Lie algebras
00:53:10 Proposition 6: for any J there exists a maximal solvable ideal
00:54:38 Proof of the proposition 6
01:01:49 Definition 5: a radical
01:02:28 Proposition 7: about radical of Lia algebra
01:03:49 Proof of the proposition 7
01:10:40 Theorem 1: rad(J) acts on irreducible complex representation of J by scalar operator
01:12:41 Proof of the theorem 1
01:27:02 Definition 6: reductive Lie algebra
01:30:10 Theorem 2: about reductive Lie algebra
01:32:10 Theorem 3: Levi's decomposition
01:34:16 Comments about the last two theorems

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