Composition algebra

English

Noun

1. Template:Lb A non-associative (not necessarily associative) Template:L, 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, Template:W, page 11,

     More precisely, $X^n\subset {\mathbb{P}}^N$ is a Severi variety if and only if ${\mathbb{P}}^N=\mathbb{P}(𝔍)$, 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 $\le 1$ (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).

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   • 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 Template:L, $(A,{}^{*},N)$, where $A$ is a nonassociative algebra, the mapping $x\to x^{*}$ is an involution, called a Template:M, and $N$ is the quadratic form $N(x)=x{x}^{*}$, called the Template:M of the algebra.
• A composition algebra may be:
  1. A Template:M if there exists some $\nu \in A:\nu \ne 0\wedge N(\nu )=0$ (called a Template:M). In this case, $N$ is called an Template:M and the algebra is said to split.
  2. A Template:M otherwise; so named because division, except by 0, is possible: the multiplicative inverse of $x$ is $x^{*}/N(x)$. In this case, $N$ is an Template:M.

Hypernyms

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Hyponyms

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Translations

• French: Template:T

Further reading

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• Division algebra on Template:W
• composition algebra on Template:W
• Division Algebra on Template:W