Уваров Ф.В. - Представления алгебр Ли. Лекции - 15. Brief introduction to Kac-Moody Lie algebras

00:00:19 Today’s plan 00:00:42 Definition of realization of matrix 00:04:13 Lemma: the connection of a realization with the dim of finite-dim vector space 00:11:34 Definition of the minimum realization 00:12:29 Proposition: the existence of the minimum realization 00:20:52 Definition of the generalized Cartan matrix 00:25:20 Construction of the certain Lie algebra 00:39:44 Lemma: weight decomposition 00:46:10 Corollary: any ideal admits weight decomposition 00:49:13 Lemma: the constructed algebra contains an ideal such intersection is trivial 00:55:00 Definition of the Kac-Moody Lie algebra 01:00:23 Define roots and root decomposition 01:04:24 Define the Weyl group 01:08:32 Proposition: The Kac-Moody Lie algebra is as the direct sum 01:22:17 Definition of the positive and negative roots 01:28:18 Definition of real and imaginary roots 01:30:19 The form (,) extends to symmetric bilinear invariant form on L(A) 01:35:36 Lemma: (,) is invariant . the action of Weyl group 01:36:24 Proposition: A root is imaginary iff (a,a) is less or equal 0 01:38:00 Proposition: All integer multiples of k are imaginary roots Курс: Представления конечномерных и бесконечномерных алгебр Ли Ссылка на плейлист: