The hit problem for the Dickson algebra



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.

Nhận xét

Bài đăng phổ biến