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Identities for the associator in alternative algebras. (English) Zbl 0997.17021

The associator in an algebra \(A\) over a field \(K\) is defined by the formula \((a,b,c)=(ab)c-a(bc)\) for any \(a\), \(b\), and \(c\) in \(A\). It is an alternative trilinear product for any alternative algebra. In the present paper, the associator is studied in a free alternative algebra, in the Cayley ternary algebra and in the ternary cross product on a four-dimensional vector space. The authors determine the identities of degree less than or equal to 7 satisfied by the associator in these three ternary algebras. Two new identities in degree 7 for alternative algebras and five new identities in degree 7 for the Cayley algebra are found. The ternary derivation identity in degree 5 introduced by Filippov for the ternary cross product is recovered. This last identity implies all the identities satisfied by the ternary cross product in degrees 5 and 7.
The results in the paper were determined by machine computation over a field with 103 elements, in order to be able to store each matrix entry in a single byte. For this reason the theorems are stated over fields of characteristic 103.

MSC:

17D05 Alternative rings
17-08 Computational methods for problems pertaining to nonassociative rings and algebras

Software:

Albert

References:

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