English Dictionary

composition algebra

名詞

(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.

日本語の意味

体上の非可換（必ずしも結合的でない）代数 A と、非退化な二次形式 N が存在し、すべての x, y ∈ A に対して N(xy)=N(x)N(y) が成立する代数。すなわち、乗法と二次形式の間に整合性（乗法的あるいは構成的性質）が認められる代数である。

In his lecture on quadratic forms, the professor explained how a composition algebra over a field is a non-associative algebra A equipped with a nondegenerate quadratic form N satisfying N(xy) = N(x)N(y) for all x, y ∈ A.

二次形式の講義で、教授は体上の非結合代数で、非退化二次形式Nを備え、任意の x, y に対して N(xy) = N(x)N(y) を満たす代数がどのようなものかを説明した。

意味（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.

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).

composition algebras （ plural ）