Triple system (original) (raw)
In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).
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| dbo:abstract | In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals). (en) 代数学における三重系または三項系（さんこうけい、英: triple system）は、ベクトル空間 V と V 上の三重積 (triple product) または三項積 (ternary product) と呼ばれる三重線型写像 との組として与えられる構造である。最も重要な例にリー三重系やジョルダン三重系があり、これらは1949年にが三項交換子 [ [u, v], w ] および三項反交換子 { u, {v, w} } に関して閉じている結合代数の部分空間を研究するために導入した。特に、任意のリー環はリー三重系を定め、任意のはジョルダン三重系を定める。これらの概念は、（特におよびその一般化であるとその非コンパクト双対）の理論において重要である。 (ja) |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=6Zl8CwAAQBAJ&pg=PR1 http://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf https://books.google.com/books%3Fid=a9KFAwAAQBAJ http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf%7Curl-status=dead%7Carchive-url=https:/web.archive.org/web/20160303234008/http:/molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf%7Carchive-date=2016-03-03 https://archive.org/details/symmetricspaces0000loos_k2a0/mode/2up http://www.emis.de/journals/JLT/vol.12_no.2/9.html%7Ctitle=Moore-Penrose |
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| dbp:first | Noriaki (en) |
| dbp:id | Jordan_triple_system (en) Lie_triple_system (en) |
| dbp:last | Kamiya (en) |
| dbp:title | Jordan triple system (en) Lie triple system (en) |
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| dcterms:subject | dbc:Representation_theory |
| rdfs:comment | In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals). (en) 代数学における三重系または三項系（さんこうけい、英: triple system）は、ベクトル空間 V と V 上の三重積 (triple product) または三項積 (ternary product) と呼ばれる三重線型写像 との組として与えられる構造である。最も重要な例にリー三重系やジョルダン三重系があり、これらは1949年にが三項交換子 [ [u, v], w ] および三項反交換子 { u, {v, w} } に関して閉じている結合代数の部分空間を研究するために導入した。特に、任意のリー環はリー三重系を定め、任意のはジョルダン三重系を定める。これらの概念は、（特におよびその一般化であるとその非コンパクト双対）の理論において重要である。 (ja) |
| rdfs:label | 三項系 (ja) Triple system (en) |
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