"composition algebra" meaning in English

{
  "forms": [
    {
      "form": "composition algebras",
      "tags": [
        "plural"
      ]
    }
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      "expansion": "composition algebra (plural composition algebras)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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          "kind": "other",
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          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
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            "Fundamental"
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        }
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      "examples": [
        {
          "text": "1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,\nMore precisely, Xⁿ⊂ℙᴺ is a Severi variety if and only if ℙᴺ=ℙ(𝔍), where 𝔍 is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra 𝔄, and X corresponds to the cone of Hermitian matrices of rank <1 (in that case SX corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X is a Severi variety if and only if X is the “Veronese surface” over one of the composition algebras over the field K (Theorem 4.9)."
        },
        {
          "ref": "1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 464",
          "text": "We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of #x5C;mathsf#x7B;Comp#x7D;#x5F;m consisting of symmetric composition algebras by #x5C;mathsf#x7B;Scomp#x7D;#x5F;m.",
          "type": "quotation"
        },
        {
          "text": "2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 150,\nAt least in the split cases, this is a construction that depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 x 3-hermitian matrices over a unital composition algebra."
        }
      ],
      "glosses": [
        "A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A."
      ],
      "hypernyms": [
        {
          "word": "non-associative algebra"
        }
      ],
      "hyponyms": [
        {
          "word": "division algebra"
        },
        {
          "word": "Hurwitz algebra"
        },
        {
          "word": "split algebra"
        }
      ],
      "id": "en-composition_algebra-en-noun-xoQ8XDVG",
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          "algebra"
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          "non-associative",
          "non-associative algebra"
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          "algebra",
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        ],
        [
          "nondegenerate",
          "nondegenerate"
        ],
        [
          "quadratic form",
          "quadratic form"
        ]
      ],
      "raw_glosses": [
        "(algebra) A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "type of nonassociative algebra",
          "tags": [
            "feminine"
          ],
          "word": "algèbre de composition"
        }
      ],
      "wikipedia": [
        "composition algebra"
      ]
    }
  ],
  "word": "composition algebra"
}

{
  "forms": [
    {
      "form": "composition algebras",
      "tags": [
        "plural"
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    }
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  "head_templates": [
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      "expansion": "composition algebra (plural composition algebras)",
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  "hypernyms": [
    {
      "word": "non-associative algebra"
    }
  ],
  "hyponyms": [
    {
      "word": "division algebra"
    },
    {
      "word": "Hurwitz algebra"
    },
    {
      "word": "split algebra"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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        "en:Algebra"
      ],
      "examples": [
        {
          "text": "1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,\nMore precisely, Xⁿ⊂ℙᴺ is a Severi variety if and only if ℙᴺ=ℙ(𝔍), where 𝔍 is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra 𝔄, and X corresponds to the cone of Hermitian matrices of rank <1 (in that case SX corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X is a Severi variety if and only if X is the “Veronese surface” over one of the composition algebras over the field K (Theorem 4.9)."
        },
        {
          "ref": "1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 464",
          "text": "We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of #x5C;mathsf#x7B;Comp#x7D;#x5F;m consisting of symmetric composition algebras by #x5C;mathsf#x7B;Scomp#x7D;#x5F;m.",
          "type": "quotation"
        },
        {
          "text": "2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 150,\nAt least in the split cases, this is a construction that depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 x 3-hermitian matrices over a unital composition algebra."
        }
      ],
      "glosses": [
        "A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A."
      ],
      "links": [
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          "non-associative",
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          "algebra",
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        ],
        [
          "nondegenerate",
          "nondegenerate"
        ],
        [
          "quadratic form",
          "quadratic form"
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      "raw_glosses": [
        "(algebra) A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "composition algebra"
      ]
    }
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  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "type of nonassociative algebra",
      "tags": [
        "feminine"
      ],
      "word": "algèbre de composition"
    }
  ],
  "word": "composition algebra"
}