Identities for algebras obtained from the Cayley-Dickson process

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2001-01-01
Authors
Bremner, Murray
Hentzel, Irvin
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Abstract

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.

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This article is published as Bremner, Murray, and Irvin Hentzel. "Identities for algebras obtained from the Cayley-Dickson process." Communications in Algebra 29, no. 8 (2001): 3523-3534. doi: 10.1081/AGB-100105036. Posted with permission.

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