Vol. 20, No. 3, 1967

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On generalized Cayley-Dickson algebras

Robert Bruce Brown

Vol. 20 (1967), No. 3, 415–422

Among those algebras whose multiplication does not satisfy the associative law is a particular family of noncommutative Jordan algebras, the generalized Cayley-Dickson algebras. These are certain central simple algebras whose dimensions are all powers of two. Most of this paper is concerned with giving the classification up to isomorphism of those of dimensions 16, 32, and 64 and determining the automorphism groups. In addition to this some generalized Cayley-Dickson division algebras are constructed. Precise criteria for when the 16-dimensional algebras are division algebras are formulated and applied to algebras over some common fields. For higher dimensions no such criteria are given. However, specific examples of division algebras for each dimension 2t are constructed over power-series fields.

Mathematical Subject Classification
Primary: 17.40
Received: 23 January 1966
Published: 1 March 1967
Robert Bruce Brown