Shestakov, Ivan; Zhukavets, Natalia Skew-symmetric identities of octonions. (English) Zbl 1241.17033 J. Pure Appl. Algebra 213, No. 4, 479-492 (2009). 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. Reviewer: Alberto Elduque (Zaragoza) Cited in 1 ReviewCited in 6 Documents MSC: 17D05 Alternative rings 17A45 Quadratic algebras (but not quadratic Jordan algebras) 17A70 Superalgebras 17A75 Composition algebras Keywords:octonions; skew-symmetric identities; quadratic alternative algebra; free superalgebra. × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Elduque, A., Quadratic alternative algebras, J. Math. Phys., 31, 1, 1-5 (1990) · Zbl 0716.17002 [2] Hentzel, I. R.; Peresi, L. A., Degree three, four, and five identities of quadratic algebras, J. Algebra, 206, 1-16 (1998) · Zbl 0915.17003 [3] Hentzel, I. R.; Peresi, L. A., Identities of Cayley-Dickson algebras, J. Algebra, 188, 292-309 (1997) · Zbl 0890.17001 [4] Il’tyakov, A. V., The speciality of the ideals of identities of some simple nonassociative algebras, Algebra and Logic, 24, 327-351 (1985), (in Russian) · Zbl 0586.17011 [5] Isaev, I. M., Identities of a finite Cayley-Dickson algebra, Algebra and Logic, 23, 282-289 (1984) · Zbl 0598.17013 [6] Osborn, J. M., Quadratic division algebras, Trans. Amer. Math. Soc., 105, 202-221 (1962) · Zbl 0136.30303 [7] Racine, M. L., Minimal identities of octonion algebras, J. Algebra, 115, 251-260 (1988) · Zbl 0651.17012 [8] Razmyslov, Ju. P., Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR, Izv., 8, 727-760 (1974), English transl.: Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974) 723-756 · Zbl 0311.16016 [9] Shestakov, I. P., Free Malcev superalgebra on one odd generator, J. Algebra Appl., 2, 4, 451-461 (2003) · Zbl 1050.17026 [10] Shestakov, I.; Zhukavets, N., The Malcev Poisson superalgebra of the free Malcev superalgebra on one odd generator, J. Algebra Appl., 5, 4, 521-535 (2006) · Zbl 1161.17325 [11] Shestakov, I.; Zhukavets, N., Speciality of Malcev superalgebras on one odd generator, J. Algebra, 301, 2, 587-600 (2006) · Zbl 1162.17028 [12] Shestakov, I.; Zhukavets, N., The free alternative superalgebra on one odd generator, Internat. J. Algebra Comp., 17, 5-6, 1215-1247 (2007) · Zbl 1205.17033 [13] Zhevlakov, K. A.; Slinko, A. M.; Shestakov, I. P.; Shirshov, A. I., Rings that are Nearly Associative (1978), Nauka: Nauka Moscow, English transl.: Academic Press, NY, 1982 · Zbl 0445.17001 [14] Zhukavets, N.; Shestakov, I., The free Malcev superalgebra on one odd generator and related superalgebras, Fundamental and Applied Mathematics, 10, 4, 97-106 (2004), (in Russian); English transl.: J. Math. Sci. (NY) 140 (2) (2007) 243-249 · Zbl 1072.17016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.