composition algebra

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composition algebra

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Noun

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, y ∈ A.
   • 1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,

     More precisely, ${\displaystyle X^{n}\subset \mathbb {P} ^{N}}$ is a Severi variety if and only if ${\displaystyle \mathbb {P} ^{N}=\mathbb {P} ({\mathfrak {J}})}$, where ${\displaystyle {\mathfrak {J}}}$ is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra ${\displaystyle {\mathfrak {A}}}$, and ${\displaystyle X}$ corresponds to the cone of Hermitian matrices of rank ${\displaystyle \leq 1}$ (in that case ${\displaystyle SX}$ corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, ${\displaystyle X}$ is a Severi variety if and only if ${\displaystyle X}$ is the “Veronese surface” over one of the composition algebras over the field ${\displaystyle K}$ (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 ${\displaystyle {\mathsf {Comp}}_{m}}$ consisting of symmetric composition algebras by ${\displaystyle {\mathsf {Scomp}}_{m}}$.

   • 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

• Formally, a tuple, ${\displaystyle (A,\ ^{*},N)}$, where ${\displaystyle A}$ is a nonassociative algebra, the mapping ${\displaystyle x\to x^{*}}$ is an involution, called a conjugation, and ${\displaystyle N}$ is the quadratic form ${\displaystyle N(x)=xx^{*}\!\!}$, called the norm of the algebra.
• A composition algebra may be:
  1. A split algebra if there exists some ${\displaystyle \nu \in A:\nu \neq 0\land N(\nu )=0}$ (called a null vector). In this case, ${\displaystyle N}$ 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 ${\displaystyle x}$ is ${\displaystyle \textstyle x^{*}/N(x)}$. In this case, ${\displaystyle N}$ is an anisotropic quadratic form.

Hypernyms

• non-associative algebra

Hyponyms

• division algebra
• Hurwitz algebra
• split algebra

Translations

Further reading

• Division algebra on Wikipedia.Wikipedia
• Cayley–Dickson construction on Wikipedia.Wikipedia
• Freudenthal magic square on Wikipedia.Wikipedia
• Hurwitz's theorem (composition algebras) on Wikipedia.Wikipedia
• Null vector on Wikipedia.Wikipedia
• Quadratic form on Wikipedia.Wikipedia
  • Isotropic quadratic form on Wikipedia.Wikipedia
• Division algebra on Encyclopedia of Mathematics
• composition algebra on nLab
• Division Algebra on Wolfram MathWorld