Английская Википедия:Composition algebra

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон:Short description Шаблон:Algebraic structures In mathematics, a composition algebra Шаблон:Mvar over a field Шаблон:Mvar is a not necessarily associative algebra over Шаблон:Mvar together with a nondegenerate quadratic form Шаблон:Mvar that satisfies

<math>N(xy) = N(x)N(y)</math>

for all Шаблон:Mvar and Шаблон:Mvar in Шаблон:Mvar.

A composition algebra includes an involution called a conjugation: <math>x \mapsto x^*.</math> The quadratic form <math>N(x) = x x^*</math> is called the norm of the algebra.

A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector. When x is not a null vector, the multiplicative inverse of x is Шаблон:Nowrap When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".

Structure theorem

Every unital composition algebra over a field Шаблон:Mvar can be obtained by repeated application of the Cayley–Dickson construction starting from Шаблон:Mvar (if the characteristic of Шаблон:Mvar is different from Шаблон:Math) or a 2-dimensional composition subalgebra (if Шаблон:Math).  The possible dimensions of a composition algebra are Шаблон:Math, Шаблон:Math, Шаблон:Math, and Шаблон:Math.

• 1-dimensional composition algebras only exist when Шаблон:Math.
• Composition algebras of dimension 1 and 2 are commutative and associative.
• Composition algebras of dimension 2 are either quadratic field extensions of Шаблон:Mvar or isomorphic to Шаблон:Math.
• Composition algebras of dimension 4 are called quaternion algebras.  They are associative but not commutative.
• Composition algebras of dimension 8 are called octonion algebras.  They are neither associative nor commutative.

For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.

Every composition algebra is an alternative algebra.

Using the doubled form ( _ : _ ): A × A → K by <math>(a:b) = n(a+b) - n(a) - n(b),</math> then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.

Instances and usage

When the field Шаблон:Mvar is taken to be complex numbers Шаблон:Math and the quadratic form Шаблон:Math, then four composition algebras over Шаблон:Math are Шаблон:Math, the bicomplex numbers, the biquaternions (isomorphic to the Шаблон:Gaps complex matrix ring Шаблон:Math), and the bioctonions Шаблон:Math, which are also called complex octonions.

The matrix ring Шаблон:Math has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra.

The squaring function Шаблон:Math on the real number field forms the primordial composition algebra. When the field Шаблон:Mvar is taken to be real numbers Шаблон:Math, then there are just six other real composition algebras.Шаблон:Rp In two, four, and eight dimensions there are both a division algebra and a split algebra:

binarions: complex numbers with quadratic form Шаблон:Math and split-complex numbers with quadratic form Шаблон:Math,

quaternions and split-quaternions,

octonions and split-octonions.

Every composition algebra has an associated bilinear form B(x,y) constructed with the norm N and a polarization identity:

<math>B(x,y) \ = \ [N(x + y) - N(x) - N(y)]/2 .</math>

History

The composition of sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied. Leonhard Euler discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of quaternions.Шаблон:Rp In 1848 tessarines were described giving first light to bicomplex numbers.

About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra:

Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras...Шаблон:Rp

In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit Шаблон:Math, and for quaternions Шаблон:Math and Шаблон:Math writes a Cayley number Шаблон:Math. Denoting the quaternion conjugate by Шаблон:Math, the product of two Cayley numbers is

<math>(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e .</math>

The conjugate of a Cayley number is Шаблон:Math, and the quadratic form is Шаблон:Math, obtained by multiplying the number by its conjugate. The doubling method has come to be called the Cayley–Dickson construction.

In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras).

In 1931 Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions. Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms. Nathan Jacobson described the automorphisms of composition algebras in 1958.

The classical composition algebras over Шаблон:Math and Шаблон:Math are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.Шаблон:Rp

See also

• Freudenthal magic square
• Pfister form
• Triality

Further reading

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book