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- Complexification_(Lie_group) abstract "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.".
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- Complexification_(Lie_group) wikiPageWikiLink Algebraic_geometry_and_analytic_geometry.
- Complexification_(Lie_group) wikiPageWikiLink Algebraic_group.
- Complexification_(Lie_group) wikiPageWikiLink Armand_Borel.
- Complexification_(Lie_group) wikiPageWikiLink Borel_subgroup.
- Complexification_(Lie_group) wikiPageWikiLink Borel–Weil_theorem.
- Complexification_(Lie_group) wikiPageWikiLink Borel–de_Siebenthal_theory.
- Complexification_(Lie_group) wikiPageWikiLink Bruhat_decomposition.
- Complexification_(Lie_group) wikiPageWikiLink Cartan_decomposition.
- Complexification_(Lie_group) wikiPageWikiLink Category:Algebraic_groups.
- Complexification_(Lie_group) wikiPageWikiLink Category:Lie_algebras.
- Complexification_(Lie_group) wikiPageWikiLink Category:Lie_groups.
- Complexification_(Lie_group) wikiPageWikiLink Category:Representation_theory.
- Complexification_(Lie_group) wikiPageWikiLink Centralizer_and_normalizer.
- Complexification_(Lie_group) wikiPageWikiLink Claude_Chevalley.
- Complexification_(Lie_group) wikiPageWikiLink Coalgebra.
- Complexification_(Lie_group) wikiPageWikiLink Compact_group.
- Complexification_(Lie_group) wikiPageWikiLink Complex_Lie_group.
- Complexification_(Lie_group) wikiPageWikiLink Complex_projective_space.
- Complexification_(Lie_group) wikiPageWikiLink Complexification.
- Complexification_(Lie_group) wikiPageWikiLink Covering_group.
- Complexification_(Lie_group) wikiPageWikiLink Coxeter_element.
- Complexification_(Lie_group) wikiPageWikiLink Coxeter_group.
- Complexification_(Lie_group) wikiPageWikiLink Double_coset.
- Complexification_(Lie_group) wikiPageWikiLink Engels_theorem.
- Complexification_(Lie_group) wikiPageWikiLink Functional_calculus.
- Complexification_(Lie_group) wikiPageWikiLink General_linear_group.
- Complexification_(Lie_group) wikiPageWikiLink Graduate_Texts_in_Mathematics.
- Complexification_(Lie_group) wikiPageWikiLink Gram–Schmidt_process.
- Complexification_(Lie_group) wikiPageWikiLink Hermitian_symmetric_space.
- Complexification_(Lie_group) wikiPageWikiLink Holomorphic_vector_bundle.
- Complexification_(Lie_group) wikiPageWikiLink Hopf_algebra.
- Complexification_(Lie_group) wikiPageWikiLink Induced_representation.
- Complexification_(Lie_group) wikiPageWikiLink Invariant_theory.
- Complexification_(Lie_group) wikiPageWikiLink Irreducible_representation.
- Complexification_(Lie_group) wikiPageWikiLink Iwasawa_decomposition.
- Complexification_(Lie_group) wikiPageWikiLink Killing_form.
- Complexification_(Lie_group) wikiPageWikiLink LU_decomposition.
- Complexification_(Lie_group) wikiPageWikiLink Lie_algebra.
- Complexification_(Lie_group) wikiPageWikiLink Lie_group.
- Complexification_(Lie_group) wikiPageWikiLink Mathematics.
- Complexification_(Lie_group) wikiPageWikiLink Matrix_coefficient.
- Complexification_(Lie_group) wikiPageWikiLink Minor_(linear_algebra).
- Complexification_(Lie_group) wikiPageWikiLink Nilpotent_group.
- Complexification_(Lie_group) wikiPageWikiLink Orbit.
- Complexification_(Lie_group) wikiPageWikiLink Polar_decomposition.
- Complexification_(Lie_group) wikiPageWikiLink Real_form_(Lie_theory).
- Complexification_(Lie_group) wikiPageWikiLink Representative_function.
- Complexification_(Lie_group) wikiPageWikiLink Robert_Steinberg.
- Complexification_(Lie_group) wikiPageWikiLink Section_(fiber_bundle).
- Complexification_(Lie_group) wikiPageWikiLink Semidirect_product.
- Complexification_(Lie_group) wikiPageWikiLink Tohoku_Mathematical_Journal.
- Complexification_(Lie_group) wikiPageWikiLink Triangular_matrix.
- Complexification_(Lie_group) wikiPageWikiLink Unitary_group.
- Complexification_(Lie_group) wikiPageWikiLink Unitary_representation.
- Complexification_(Lie_group) wikiPageWikiLink Weight_(representation_theory).
- Complexification_(Lie_group) wikiPageWikiLink Weyl_group.
- Complexification_(Lie_group) wikiPageWikiLinkText "Complexification (Lie group)".
- Complexification_(Lie_group) wikiPageWikiLinkText "Complexification (Lie group)#Complex structures on homogeneous spaces".
- Complexification_(Lie_group) wikiPageWikiLinkText "Gauss decomposition".
- Complexification_(Lie_group) wikiPageWikiLinkText "complexification".
- Complexification_(Lie_group) wikiPageWikiLinkText "complexify the Lie algebra".
- Complexification_(Lie_group) wikiPageWikiLinkText "real form".
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- Complexification_(Lie_group) subject Category:Algebraic_groups.
- Complexification_(Lie_group) subject Category:Lie_algebras.
- Complexification_(Lie_group) subject Category:Lie_groups.
- Complexification_(Lie_group) subject Category:Representation_theory.
- Complexification_(Lie_group) type Group.
- Complexification_(Lie_group) type Algebra.
- Complexification_(Lie_group) type Field.
- Complexification_(Lie_group) type Group.
- Complexification_(Lie_group) type Variety.
- Complexification_(Lie_group) comment "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.".
- Complexification_(Lie_group) label "Complexification (Lie group)".
- Complexification_(Lie_group) sameAs Q17005573.
- Complexification_(Lie_group) sameAs m.0r8k2z_.
- Complexification_(Lie_group) sameAs Q17005573.
- Complexification_(Lie_group) wasDerivedFrom Complexification_(Lie_group)?oldid=686599342.
- Complexification_(Lie_group) isPrimaryTopicOf Complexification_(Lie_group).