✦ LIBER ✦

Identities for the Associator in Alternative Algebras

✍ Scribed by Murray Bremner; Irvin Hentzel

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 273 KB
Volume: 33
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.2001.0510

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✦ Synopsis

The associator is an alternating trilinear product for any alternative algebra. We study this trilinear product in three related algebras: the associator in a free alternative algebra, the associator in the Cayley algebra, and the ternary cross product on four-dimensional space. This last example is isomorphic to the ternary subalgebra of the Cayley algebra which is spanned by the non-quaternion basis elements. We determine the identities of degree ≤ 7 satisfied by these three ternary algebras. We discover two new identities in degree 7 satisfied by the associator in every alternative algebra and five new identities in degree 7 satisfied by the associator in the Cayley algebra. For the ternary cross product we recover the ternary derivation identity in degree 5 introduced by Filippov.

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