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 Ссылка на плейлист: #мгу #мехмат #уваров #группыли #алгебрыли