Vol. 64, No. 2, 1976

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Bonded quadratic division algebras

R. A. Czerwinski

Vol. 64 (1976), No. 2, 341–351

Osborn has shown that any quadratic algebra over a field of characteristic not two can be decomposed into a copy of the field and a skew-commutative algebra with a bilinear form. For any nonassociative algebra G over a field of characteristic not two, Albert and Oehmke have defined an algebra over the same vector space, which is bonded to G by a linear transformation T. In this paper this process is specialized to the class 𝒜 of finite dimensional quadratic algebras A over fields of characteristic not two, which define a symmetric, nondegenerate bilinear form, to obtain quadratic algebras B(A,T) bonded to A. In the main results T will be defined as a linear transformation on the skew-commutative algebra V defined by Osborn’s decomposition of A. An algebra in 𝒜 is called a division algebra if A0 and the equations ax = b and ya = b, where a0 and b are elements in A, have unique solutions for x and y in A. Consequently, a finite dimensional algebra A0 is a division algebra if and only if A has no divisors of zero. A basis for V is said to be orthogonal, if it is orthogonal with respect to the above mentioned bilinear form. An algebra in 𝒜 is weakly flexible if the i-th component of the skew-commutative product of the i-th and j-th members of each orthogonal basis of V is 0. If 𝒟(𝒜) denotes the class of division algebras in 𝒜 and I denotes the identity transformation on V , then the main results are: (1) A ∈𝒟(𝒜), T nonsingular and B(A,T) flexible imply B(A,T) ∈𝒟(𝒜), (2) if A ∈𝒟(𝒜) and A is weakly flexible, then B(A,T) is weakly flexible if and only if T = δI for δ a scalar, and (3) if A is a Cayley-Dickson algebra in 𝒟(𝒜), then B(A,T) is a Cayley-Dickson algebra in 𝒟(𝒜) if and only if T = ±I. Finally, a class of nonflexible quadratic division algebras bonded to Cayley-Dickson division algebras will be exhibited.

Mathematical Subject Classification 2000
Primary: 17A99
Received: 5 November 1975
Published: 1 June 1976
R. A. Czerwinski