In
mathematics, the Cayley–Dickson construction, named after Arthur Cayley and
Leonard Eugene Dickson, produces a sequence of algebras over the field of real
numbers, each with twice the dimension of the previous one. The algebras
produced by this process are known as Cayley–Dickson algebras. They are useful
composition algebras frequently applied in mathematical physics.
The
Cayley–Dickson construction defines a new algebra similar to the direct sum of
an algebra with itself, with multiplication defined in a specific way
(different from the multiplication provided by the genuine direct sum) and an
involution known as conjugation. The product of an element and its conjugate
(or sometimes the square root of this product) is called the norm.
The
symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly
applied: first losing order, then commutativity of multiplication,
associativity of multiplication, and next alternativity.
More
generally, the Cayley–Dickson construction takes any algebra with involution to
another algebra with involution of twice the dimension.[1]:45
Let
the mod 2 Steenrod algebra, A, and the general linear group, GL(k; F-2), act on
P-k := F-2[x(1),...,x(k)] with \x(i)\ = 1 in the usual manner. We prove the
conjecture of the first-named author in Spherical classes and the algebraic
transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every
element of positive degree in the Dickson algebra D-k := (P-k)(GL(k; F2)) is
A-decomposable in P-k for arbitrary k>2. This conjecture was shown to be
equivalent to a weak algebraic version of the classical conjecture on spherical
classes, which states that the only spherical classes in Q(0)S(0) are the
elements of Hopf invariant one and those of Kervaire invariant one.
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