About: Split-octonion

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In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0).

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rdfs:label	Octonion déployé (fr) 分解型八元数 (ja) Split-octonion (en)
rdfs:comment	En mathématiques, les octonions déployés ou octonions fendus sont une extension non associative des quaternions (ou des coquaternions). Ils diffèrent des octonions par la signature de la forme quadratique : les octonions déployés ont une signature mixte (4,4) alors que les octonions ont une signature définie positive (8,0). (fr) 数学における分解型八元数（ぶんかいがたはちげんすう、英: split-octonion）の全体は、実八次元の分配多元環を成す。通常の八元数とは異なり、非可逆な非零元を含む。またその計量二次形式（（二次の）ノルム）の符号数も異なり、通常の八元数のが正定値符号数 (8, 0) を持つのに対して、分解型八元数のは分解型符号数 (4, 4) を持つ。八元数全体と分解型八元数全体の二者が、同型を除いて可能な実数体 ℝ 上の一般八元数環の全てを尽くす。任意の体 F 上でも対応する分解型の八元数環を考えることができる。 (ja) In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0). (en)
foaf:depiction
dcterms:subject	Composition algebras Octonions
Wikipage page ID	3861649 (xsd:integer)
Wikipage revision ID	994676253 (xsd:integer)
Link from a Wikipage to another Wikipage	Pseudo-Euclidean space Moufang loop Moufang identities Nonassociative Basis (linear algebra) Determinant Multiplicative inverse Quaternions Cross product Mathematics Matrix (mathematics) Max August Zorn Mnemonic Levi-Civita symbol Signature (quadratic form) Composition algebra Cayley–Dickson construction G2 (mathematics) Linear combination Alternative algebra Field (mathematics) Isotropic quadratic form Quadratic form Simple Lie group Artin's theorem Composition algebras Octonions Dirac equation Dot product Split-quaternion Kronecker delta Octonion Real number Supersymmetric quantum mechanics Octonion algebra Multiplication table Real form (Lie theory) Il Nuovo Cimento B
sameAs	Split-octonion Split-octonion Split-octonion Split-octonion Split-octonion Split-octonion
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Doi dbt:Reflist dbt:Number_systems
thumbnail	wiki-commons:Special:FilePath/SplitFanoPlane.svg?width=300
has abstract	In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two octonion algebras over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field. (en) En mathématiques, les octonions déployés ou octonions fendus sont une extension non associative des quaternions (ou des coquaternions). Ils diffèrent des octonions par la signature de la forme quadratique : les octonions déployés ont une signature mixte (4,4) alors que les octonions ont une signature définie positive (8,0). (fr) 数学における分解型八元数（ぶんかいがたはちげんすう、英: split-octonion）の全体は、実八次元の分配多元環を成す。通常の八元数とは異なり、非可逆な非零元を含む。またその計量二次形式（（二次の）ノルム）の符号数も異なり、通常の八元数のが正定値符号数 (8, 0) を持つのに対して、分解型八元数のは分解型符号数 (4, 4) を持つ。八元数全体と分解型八元数全体の二者が、同型を除いて可能な実数体 ℝ 上の一般八元数環の全てを尽くす。任意の体 F 上でも対応する分解型の八元数環を考えることができる。 (ja)
prov:wasDerivedFrom	wikipedia-en:Split-octonion?oldid=994676253&ns=0
page length (characters) of wiki page	11595 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Split-octonion
is Link from a Wikipage to another Wikipage of	Moufang loop List of irreducible Tits indices Max August Zorn Nilpotent Composition algebra Cayley–Dickson construction Hypercomplex number Split-biquaternion Split-complex number Split-octonions Split-quaternion Octonion Octonion algebra Zorn's vector-matrix algebra Split-octonian Split octonions
is Wikipage redirect of	Split-octonions Zorn's vector-matrix algebra Split-octonian Split octonions
is known for of	Max August Zorn
is foaf:primaryTopic of	wikipedia-en:Split-octonion

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