The Cayley–Dickson double of a ring with involution #
For a possibly nonassociative ring A with an involution star, the Cayley–Dickson double
CayleyDickson A is the set of ordered pairs ⟨a, b⟩ of elements of A, added coordinatewise
and multiplied by
⟨a, b⟩ * ⟨c, d⟩ = ⟨a * c - star d * b, d * a + b * star c⟩,
with unit ⟨1, 0⟩ and the involution star ⟨a, b⟩ = ⟨star a, -b⟩. Doubling ℝ with its trivial
involution gives ℂ, doubling ℂ gives the quaternions, and doubling the quaternions ℍ[ℝ]
gives the octonions, the eight-dimensional real division algebra; the identities of the double of
the quaternions are proved in TauCeti.Algebra.CayleyDickson.Quaternion.
The double of an associative ring is in general no longer associative, so it is a NonAssocRing
and nothing more. This file builds that structure for every possibly nonassociative ring with
involution: the additive and module structures, transported from A × A, the multiplication with
its unit and integer casts,
the involution, which reverses products, so that the double is again a ring with involution, and
the two scalar towers for scalars acting on A compatibly with its product (and, for
SMulCommClass, with its involution).
Main definitions #
TauCeti.CayleyDickson A: the Cayley–Dickson double ofA, the ordered pairs⟨fst, snd⟩.TauCeti.CayleyDickson.linearEquivProd: the coordinates of an element of the double, as a linear isomorphism withA × A.
Main results #
- The double is a
NonAssocRingwith the product above, aStarRingwith the involutionstar ⟨a, b⟩ = ⟨star a, -b⟩, which reverses products, and a module over every ring of scalars acting onA, with the two scalar towers when the action is compatible with the product ofA(and, forSMulCommClass, with its involution). TauCeti.CayleyDickson.finrank_eq: the double has twice the dimension ofA.
Implementation notes #
The double is a structure with two fields rather than the type A × A itself, so that it carries
neither the coordinatewise product nor the coordinatewise involution of a product type. Several
equivalent sign conventions for the product are in use; with the one chosen here the unit is
⟨1, 0⟩, the element ⟨0, 1⟩ squares to -1, and the involution is
star ⟨a, b⟩ = ⟨star a, -b⟩, positive on the double of ℍ[ℝ].
References #
- J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205, §2.2.
- T. A. Springer and F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer (2000), §1.5.
The Cayley–Dickson double of a ring with involution A: the ordered pairs ⟨fst, snd⟩ of
elements of A, multiplied by ⟨a, b⟩ * ⟨c, d⟩ = ⟨a * c - star d * b, d * a + b * star c⟩.
- fst : A
The first coordinate of an element of the Cayley–Dickson double.
- snd : A
The second coordinate of an element of the Cayley–Dickson double.
Instances For
The additive and module structure #
The natural-number and integer casts of the double lie in the first coordinate.
The coordinates of an element of the Cayley–Dickson double, as a linear isomorphism with
A × A, over any semiring acting on A.
Instances For
The double has twice the dimension: an element is a pair of elements of A.
The multiplication and the involution #
The Cayley–Dickson product ⟨a, b⟩ * ⟨c, d⟩ = ⟨a * c - star d * b, d * a + b * star c⟩.
The involution star ⟨a, b⟩ = ⟨star a, -b⟩ of the Cayley–Dickson double.
The involution of the double reverses products, as the involution of A does, so the double
is again a ring with involution.
Scalars acting on A compatibly with its product and involution commute with left
multiplication in the double.
Scalars acting on A compatibly with its product associate with the product of the double.