symmetric monoidal (∞,1)-category of spectra
The Cayley-Dickson construction or Cayley-Dickson double (Dickson 1919, (6)) takes a real star-algebra to a new real star-algebra whose elements are pairs of elements of , in generalization of how the complex numbers arise as a doubling of the real numbers.
When iteratively applied to the real numbers, regarded as a star-algebra with trivial involution, the Cayley-Dickson construction yields, consecutively, the complex numbers, then the quaternions, then the octonions (thus all four real normed division algebras), then the sedenions, …
Let be an possibly nonassociative star-algebra over the field of real numbers: an algebra equipped with an involution which is an antiautomorphism. (Actually, could be replaced by any commutative ring in the definitions, although some properties may depend on this ring.)
(Cayley-Dickson construction in components)
The Cayley–Dickson double of the real star-algebra is the real star-algebra
whose underlying real vector space is the direct sum ,
whose multiplication is given by
whose star-involution is given by
(Cayley-Dickson double by generators and relations)
The Cayley-Dickson double of a real star-algebra is the real star-algebra obtained by adjoining one generator to subject to the following relations:
and
for all .
(induced relations)
The relation in Def. imply the following further relations:
for all .
Definition and Definition are equivalent, in that we have an isomorphism of real star-algebras:
It is clear from Def. that for every element there is a unique pair of elements such that
This means that is a linear isomorphism of the underlying real vector spaces. Hence it only remains to check that is indeed an algebra homomorphism and that it respects the involution.
To see that is an algebra homomorphism, we multiply out and then use the relations (7) and (8) from Lemma :
Here in the last line we indeed find the component formula (1).
To see that respects the involution we use (3) from Def. and (5) from Lemma :
Here in the last line we indeed find the component formula (2).
The map is a monomorphism . If is unital with unit then is unital with unit . In the unital case, the element has the property , and we may write as (while ). For this reason, we may write in place of , at least when is unital.
Generally speaking, the double of an algebra has a nice property iff is one level nicer. For simplicity, assume that is unital (so that is a subalgebra). Since , we see that the involution on is trivial iff the involution on is trivial and further has . Since , is commutative iff is commutative and the involution in is trivial. Since , is associative iff is associative and commutative. Finally, is alternative iff is associative (and hence also alternative).
The standard example is the sequence of consecutive doubles starting with itself (with the identity map as involution); these are the Cayley–Dickson algebras: the real numbers , the complex numbers , the quaternions , the octonions (or Cayley numbers) ,
These are the real normed division algebras
The diagram on the right shows a basis of imaginary octonions obtained via Cayley-Dickson doubling from a standard basis of imaginary quaternions.
Next, the CD-double of the octonions is the sedenions , etc. , followed by further algebras which are not division algebras.
All of these algebras are power-associative, flexible, and unital, and have all inverse elements; the subalgebra with is always just .
Named after Arthur Cayley and Leonard Dickson.
The original article:
Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)
Review and introduction:
M M Postnikov, Lectures on geometry, Semester V: Lie groups and Lie algebras, Lec. 14 (russian and english editions)
John Baez, The Cayley–Dickson construction, in The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205, doi
Tevian Dray, Corinne Manogue, Section 5.1 of: The Geomety of Octonions, World Scientific 2015 (doi:10.1142/8456)
See also
More:
Daniel K. Biss, Daniel Dugger, Daniel Isaksen, Large annihilators in Cayley-Dickson algebras, Communications in Algebra 36 (2), 632-664, 2008 (arxiv:math/0511691)
Daniel K. Biss, Daniel Christensen, Daniel Dugger, Daniel Isaksen, Large annihilators in Cayley-Dickson algebras II, Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 (arxiv:math/0702075)
Daniel K. Biss, Daniel Christensen, Daniel Dugger, Daniel Isaksen, Eigentheory of Cayley-Dickson algebras, Forum Mathematicum 21(5) (2009), 833-851 (arxiv:0905.2987)
Last revised on July 21, 2020 at 17:01:32. See the history of this page for a list of all contributions to it.