
The paper under review is devoted to the classification of the multilinear skew-symmetric identities and central polynomials of the octonions over a field of characteristic \(0\). It turns out that any multilinear skew-symmetric identity of an octonion algebra over such a field is a consequence of an identity of degree \(5\) and two identities of degree \(6\), while any multilinear skew-symmetric central polynomial is a consequence of a polynomial of degree \(4\) and another of degree \(5\). Actually, the authors work in the more general setting of quadratic alternative algebras, that is, alternative algebras satisfying an analogue of the Cayley-Hamilton equation for \(2\times 2\) matrices. This setting can be extended to deal with superalgebras, with a natural definition of a supertrace with values in the center of the superalgebra. In this way, the authors study the free quadratic alternative superalgebra on one odd generator, and they compute the super-identities and central functions on one odd variable in quadratic alternative superalgebras. But the identities of the free alternative superalgebra in one odd generator are transformed in skew-symmetric multilinear identities of the free alternative algebra on a countable set of generators. The final step consists in proving that the skew-symmetric identities and central functions of the octonions coincide with those for the class of all quadratic alternative algebras.
octonions, free superalgebra., Superalgebras, Algebra and Number Theory, skew-symmetric identities, Quadratic algebras (but not quadratic Jordan algebras), Composition algebras, quadratic alternative algebra, Alternative rings
octonions, free superalgebra., Superalgebras, Algebra and Number Theory, skew-symmetric identities, Quadratic algebras (but not quadratic Jordan algebras), Composition algebras, quadratic alternative algebra, Alternative rings
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