Difference between revisions of "Cayley-Dickson algebra"
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An alternative -dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. Quaternion and Alternative rings and algebras). The latter starts out from a given algebra
to construct a new algebra
(of twice the dimension of
) and is a generalization of the doubling process (see Hypercomplex number). Namely, let
be an algebra with a unit 1 over a field
, let
be some non-zero element of
, and let
be an
-linear mapping which is an involution, and such that
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The formula
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now defines a multiplication operation on the direct sum of linear spaces , relative to which
is an algebra. The algebra
may be imbedded in
as a subalgebra:
, and the involution
extends to an involution in
:
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Moreover,
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The extension of to
can be repeated resulting in an ascending chain of algebras
; the parameter
need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra
with basis
, multiplication table
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and involution ,
, the first application of the process yields an algebra
of generalized quaternions (an associative algebra of dimension 4), and the second — an
-dimensional algebra, known as a Cayley–Dickson algebra.
Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over ; conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form
in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:
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This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form (the norm of
) does not represent the zero in
. If
is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis
with the following multiplication table:'
<tbody> </tbody>
|
where ,
, and the involution is defined by the conditions
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This algebra is denoted by . The algebras
and
are isomorphic if and only if their quadratic forms
are equivalent. If
represents zero, the corresponding Cayley–Dickson algebra is isomorphic to
, which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices
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where ,
, with
a three-dimensional space over
with the usual definition of the scalar product
and vector product
. Matrix multiplication is defined by
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If is the real field, then
is the algebra of Cayley numbers (a division algebra). Any Cayley–Dickson algebra over
is isomorphic to either
or
.
The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [1], [2]).
Let be an alternative ring whose associative-commutative centre
is distinct from zero and does not contain zero divisors; let
be the field of fractions of
. Then there is a natural imbedding
. If
is a Cayley–Dickson algebra over
, then
is known as a Cayley–Dickson ring.
References
[1] | L.E. Dickson, "Linear algebras" , Cambridge Univ. Press (1930) |
[2] | R.D. Schafer, "An introduction to nonassociative algebras" , Acad. Press (1966) |
[3] | K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian) |
Cayley-Dickson algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Dickson_algebra&oldid=11507