composition algebra

== English == === Noun === composition algebra (plural composition algebras) (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, X n ⊂ P N {\displaystyle X^{n}\subset \mathbb {P} ^{N}} is a Severi variety if and only if P N = P ( J ) {\displaystyle \mathbb {P} ^{N}=\mathbb {P} ({\mathfrak {J}})} , where J {\displaystyle {\mathfrak {J}}} is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra A {\displaystyle {\mathfrak {A}}} , and X {\displaystyle X} corresponds to the cone of Hermitian matrices of rank ≤ 1 {\displaystyle \leq 1} (in that case S X {\displaystyle SX} corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X {\displaystyle X} is a Severi variety if and only if X {\displaystyle X} is the “Veronese surface” over one of the composition algebras over the field K {\displaystyle K} (Theorem 4.9). 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, ( A , ∗ , N ) {\displaystyle (A,\ ^{*},N)} , where A {\displaystyle A} is a nonassociative algebra, the mapping x → x ∗ {\displaystyle x\to x^{*}} is an involution, called a conjugation, and N {\displaystyle N} is the quadratic form N ( x ) = x x ∗ {\displaystyle N(x)=xx^{*}\!\!} , called the norm of the algebra. A composition algebra may be: A split algebra if there exists some ν ∈ A : ν ≠ 0 ∧ N ( ν ) = 0 {\displaystyle \nu \in A:\nu \neq 0\land N(\nu )=0} (called a null vector). In this case, N {\displaystyle N} is called an isotropic quadratic form and the algebra is said to split. A division algebra otherwise; so named because division, except by 0, is possible: the multiplicative inverse of x {\displaystyle x} is x ∗ / N ( x ) {\displaystyle \textstyle x^{*}/N(x)} . In this case, N {\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