Complex structures on homogeneous spaces | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-03

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Complexification_(Lie_group)


00:01:23 1 Universal complexification
00:01:47 1.1 Definition
00:01:58 1.2 Existence
00:02:11 1.3 Uniqueness
00:03:00 1.4 Injectivity
00:03:39 2 Chevalley complexification
00:05:17 2.1 Hopf algebra of matrix coefficients
00:05:36 2.2 Invariant theory
00:06:14 3 Decompositions in the Chevalley complexification
00:06:25 3.1 Cartan decomposition
00:07:58 3.2 Gauss decomposition
00:08:43 3.3 Bruhat decomposition
00:10:08 3.4 Iwasawa decomposition
00:10:37 4 Complex structures on homogeneous spaces
00:11:24 5 Noncompact real forms
00:11:37 5.1 Involutions of simply connected compact Lie groups
00:12:47 5.2 Conjugations on the complexification
00:18:43 5.3 Cartan decomposition in a real form
00:20:16 5.4 Iwasawa decomposition in a real form
00:21:13 6 See also
00:23:11 7 Notes
00:23:56 8 References
00:24:13 Iwasawa decomposition
00:26:48 GAN. From the properties of the Iwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in GC, which is closed. On the other hand, the dimension of the image is the same as the dimension of GC, so it is also open. So GC
00:27:49 XDY with X in N−, D in TC and Y in N+. Since h is self-adjoint, uniqueness forces Y
00:28:16 exp iT for some unique T in 𝖙. Let a
00:28:34 Y and k
00:28:51 kan.
00:28:54 Complex structures on homogeneous spaces
00:31:49 P ∩ G is given by p ∩ 𝖌. By the Iwasawa decomposition GC
00:34:02 Noncompact real forms
00:34:27 Involutions of simply connected compact Lie groups
00:39:32 Conjugations on the complexification
00:45:49 Cartan decomposition in a real form
00:49:31 u ⋅ p with u in G and p in P. Since c0 g
00:49:40 u and σp
00:51:49 1. Setting k
00:52:03 0 and hence [X,Y]
00:54:04 Iwasawa decomposition in a real form
00:54:15 See also



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SUMMARY
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In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition g = u • exp iX, where u is a unitary operator in the compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.