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- Invariant_convex_cone abstract "In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group. The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign). The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every point in the interior of the cone intersects the interior of the Weyl group invariant cone.For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones.Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii. They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series. The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual. The semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of the infinitesimal operator corresponding to an element in the maximal cone. A similar decomposition already occurs in the semigroup.The oscillator semigroup of Roger Howe concerns the special case of this theory for the real symplectic group. Historically this has been one of the most important applications and has been generalized to infinite dimensions. This article treats in detail the example of the invariant convex cone for the symplectic group and its use in the study of the symplectic Olshanskii semigroup.".
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- Invariant_convex_cone wikiPageWikiLink Adjoint_representation.
- Invariant_convex_cone wikiPageWikiLink Autonne–Takagi_factorization.
- Invariant_convex_cone wikiPageWikiLink Bertram_Kostant.
- Invariant_convex_cone wikiPageWikiLink Brouwer_fixed-point_theorem.
- Invariant_convex_cone wikiPageWikiLink Brouwers_fixed_point_theorem.
- Invariant_convex_cone wikiPageWikiLink Category:Lie_algebras.
- Invariant_convex_cone wikiPageWikiLink Category:Lie_groups.
- Invariant_convex_cone wikiPageWikiLink Category:Representation_theory.
- Invariant_convex_cone wikiPageWikiLink Category:Semigroup_theory.
- Invariant_convex_cone wikiPageWikiLink Complexification_(Lie_group).
- Invariant_convex_cone wikiPageWikiLink Contraction_(operator_theory).
- Invariant_convex_cone wikiPageWikiLink Convex_cone.
- Invariant_convex_cone wikiPageWikiLink Earle–Hamilton_fixed-point_theorem.
- Invariant_convex_cone wikiPageWikiLink Earle–Hamilton_fixed_point_theorem.
- Invariant_convex_cone wikiPageWikiLink Ernest_Vinberg.
- Invariant_convex_cone wikiPageWikiLink Haar_measure.
- Invariant_convex_cone wikiPageWikiLink Harish-Chandra.
- Invariant_convex_cone wikiPageWikiLink Hermitian_symmetric_space.
- Invariant_convex_cone wikiPageWikiLink Hilbert–Schmidt_operator.
- Invariant_convex_cone wikiPageWikiLink Holomorphic_discrete_series.
- Invariant_convex_cone wikiPageWikiLink Holomorphic_discrete_series_representation.
- Invariant_convex_cone wikiPageWikiLink Holomorphic_functional_calculus.
- Invariant_convex_cone wikiPageWikiLink Killing_form.
- Invariant_convex_cone wikiPageWikiLink Lie_algebra.
- Invariant_convex_cone wikiPageWikiLink Lie_group.
- Invariant_convex_cone wikiPageWikiLink Mathematics.
- Invariant_convex_cone wikiPageWikiLink Maximal_compact_subgroup.
- Invariant_convex_cone wikiPageWikiLink Maximal_torus.
- Invariant_convex_cone wikiPageWikiLink Oscillator_representation.
- Invariant_convex_cone wikiPageWikiLink Oscillator_semigroup.
- Invariant_convex_cone wikiPageWikiLink Poisson_bracket.
- Invariant_convex_cone wikiPageWikiLink Polar_decomposition.
- Invariant_convex_cone wikiPageWikiLink Quantum_mechanics.
- Invariant_convex_cone wikiPageWikiLink Roger_Evans_Howe.
- Invariant_convex_cone wikiPageWikiLink Symmetric_cone.
- Invariant_convex_cone wikiPageWikiLink Symmetric_matrix.
- Invariant_convex_cone wikiPageWikiLink Symplectic_group.
- Invariant_convex_cone wikiPageWikiLink Weyl_calculus.
- Invariant_convex_cone wikiPageWikiLink Weyl_group.
- Invariant_convex_cone wikiPageWikiLinkText "Invariant convex cone".
- Invariant_convex_cone hasPhotoCollection Invariant_convex_cone.
- Invariant_convex_cone wikiPageUsesTemplate Template:Citation.
- Invariant_convex_cone wikiPageUsesTemplate Template:Harvtxt.
- Invariant_convex_cone wikiPageUsesTemplate Template:Math.
- Invariant_convex_cone wikiPageUsesTemplate Template:Reflist.
- Invariant_convex_cone subject Category:Lie_algebras.
- Invariant_convex_cone subject Category:Lie_groups.
- Invariant_convex_cone subject Category:Representation_theory.
- Invariant_convex_cone subject Category:Semigroup_theory.
- Invariant_convex_cone hypernym Cone.
- Invariant_convex_cone type Mountain.
- Invariant_convex_cone type Algebra.
- Invariant_convex_cone type Field.
- Invariant_convex_cone comment "In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group.".
- Invariant_convex_cone label "Invariant convex cone".
- Invariant_convex_cone sameAs m.0swklsy.
- Invariant_convex_cone sameAs Q17098165.
- Invariant_convex_cone sameAs Q17098165.
- Invariant_convex_cone wasDerivedFrom Invariant_convex_cone?oldid=652783960.
- Invariant_convex_cone isPrimaryTopicOf Invariant_convex_cone.