About: Invariant convex cone

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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 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.

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rdf:type	yago:WikicatLieAlgebras yago:WikicatLieGroups yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Content105809192 yago:Discipline105996646 yago:Group100031264 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 mountain yago:Science105999797
rdfs:label	Invariant convex cone (en)
rdfs: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 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. (en)
dcterms:subject	Lie algebras Semigroup theory Lie groups Representation theory
Wikipage page ID	39078438 (xsd:integer)
Wikipage revision ID	1057692508 (xsd:integer)
Link from a Wikipage to another Wikipage	Quantum mechanics Roger Evans Howe Bertram Kostant Holomorphic function Infinitesimal transformation Lie group Lie algebras Semigroup theory Complexification (Lie group) Mathematics Generating set of a group Contraction (operator theory) Convex cone Oscillator representation Lie algebra Closed set Compact group Symplectic group Maximal compact subgroup Center (group theory) Dual cone and polar cone Haar measure Semigroup Adjoint representation Lie groups Ernest Vinberg Hilbert–Schmidt operator Harish-Chandra Representation theory Killing form Symmetric cone Hermitian symmetric space Holomorphic functional calculus Maximal torus Poisson bracket Polar decomposition Circle group Group isomorphism Inner automorphism Interior (topology) Simple Lie algebra Weyl group Autonne–Takagi factorization Brouwer's fixed point theorem Oscillator semigroup Earle–Hamilton fixed point theorem Weyl calculus Holomorphic discrete series dbr:Group_actin
Link from a Wikipage to an external page	http://www.heldermann-verlag.de/jlt/jlt04/LAWPL.PDF https://archive.org/details/mathematicalheri0000symp/page/61
sameAs	Invariant convex cone Invariant convex cone Invariant convex cone Invariant convex cone
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Math dbt:Reflist
has 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 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. (en)
gold:hypernym	Cone
prov:wasDerivedFrom	wikipedia-en:Invariant_convex_cone?oldid=1057692508&ns=0
page length (characters) of wiki page	25428 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Invariant_convex_cone
is Link from a Wikipage to another Wikipage of	Glossary of Lie groups and Lie algebras Oscillator representation Hermitian symmetric space
is foaf:primaryTopic of	wikipedia-en:Invariant_convex_cone

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