composition algebra

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English

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Noun

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composition algebra (plural composition algebras)

  1. (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, yA.
    • 1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,
      More precisely, is a Severi variety if and only if , where is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra , and corresponds to the cone of Hermitian matrices of rank (in that case corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, is a Severi variety if and only if is the “Veronese surface” over one of the composition algebras over the field (Theorem 4.9).
    • 1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 464:
      We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of consisting of symmetric composition algebras by .
    • 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,
      At 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.

Usage notes

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  • Formally, a tuple, , where is a nonassociative algebra, the mapping is an involution, called a conjugation, and is the quadratic form , called the norm of the algebra.
  • A composition algebra may be:
    1. A split algebra if there exists some (called a null vector). In this case, is called an isotropic quadratic form and the algebra is said to split.
    2. A division algebra otherwise; so named because division, except by 0, is possible: the multiplicative inverse of is . In this case, is an anisotropic quadratic form.

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