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In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by . The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicat

Attributes	Values
rdf:type	Thing yago:WikicatLieAlgebras yago:WikicatLieGroups yago:WikicatNon-associativeAlgebras 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	Symmetrischer Kegel (de) Symmetric cone (en)
rdfs:comment	In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by . The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicat (en)
rdfs:seeAlso	Peirce decomposition Cayley transform Formally real Jordan algebra
foaf:depiction
dcterms:subject	Lie algebras Several complex variables Lie groups Convex geometry Non-associative algebras
Wikipage page ID	39156141 (xsd:integer)
Wikipage revision ID	1071197083 (xsd:integer)
Link from a Wikipage to another Wikipage	Quadratic Jordan algebra Quaternion Montel's theorem Convex span Hurwitz's theorem (composition algebras) Jordan algebra University of Göttingen Iwasawa decomposition Lie group Sylvester's law of inertia Peirce decomposition Quadratic representation Lie algebras Complexification (Lie group) Cone Several complex variables Mathematics Operator norm Operator theory Simple group Mir Publishers Möbius transformation Convex cone Convex hull Bergman metric Lexicographic order Commutator subgroup Complex manifold Composition algebra Polarization of an algebraic form Matrix unit Maximal compact subgroup Maximum modulus principle Cauchy's integral formula Cayley transform Center (geometry) Tube domain Adolf Hurwitz Lie groups Alternative algebra Euler operator (digital geometry) Extreme point Gram matrix Jordan operator algebra Kantor–Koecher–Tits construction Koecher–Vinberg theorem Positive operator Rational function Recurrence relation Riemannian symmetric space Henri Cartan Hurwitz's theorem (normed division algebras) Riemann sphere Convex geometry Albert algebra Non-associative algebras Killing form Biholomorphism Hermitian symmetric space Associator Polar decomposition Spectral theorem Idempotent Inner product space Krein–Milman theorem

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