Meaning of composition algebra | Babel Free

Definitions

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.

Equivalents

Examples

“1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11, More 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).”

“We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of #92;mathsf#123;Comp#125;#95;m consisting of symmetric composition algebras by #92;mathsf#123;Scomp#125;#95;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.”

CEFR level