Bremner, Murray R.; Peresi, Luiz A. Nonhomogeneous subalgebras of Lie and special Jordan superalgebras. (English) Zbl 1196.17025 J. Algebra 322, No. 6, 2000-2026 (2009). The authors work with Lie and special Jordan superalgebras. They consider the underlying algebra, that is, ignoring the grading and regarding superalgebras as ordinary algebras. Volichenko investigated polynomial identities satisfied by all nonhomogeneous subalgebras of Lie superalgebras (Volichenko algebras). He found a set of identities which he believed would imply any other identity in degree \(\leq 5\). However, A. A. Baranov [Sib. Math. J. 36, No. 5, 859–868 (1995); translation from Sib. Mat. Zh. 36, No. 5, 998–1009 (1995; Zbl 0860.17007)] found an identity in degree \(5\) which did not follow from Volichenko’s identities. Furthermore, Baranov proved that the set of identities of Volichenko together with his new identity imply then all identities in degree \(\leq 6\). There are also a number of publications by Leites and Serganova (1990, 1992 and 2001) dealing with the classification problem of simple finite-dimensional Volichenko algebras.In this work the authors verify that the identities of Volichenko and Baranov in degree \(\leq 5\) imply all the identities in degree \(\leq 6\) for Volichenko algebras. They also prove that there are no new identities in degree \(7\). They find a new identity that is equivalent to the two previously known identities in degree \(5\) and show that every identity in degree \(\leq 7\) follows from \(11\) irreducible identities in degree \(\leq 5\).Since the authors’ methodology is rather independent of the nature of the algebra under study, they tackle also the Jordan case. Jordan superalgebras had not been considered from this viewpoint in the literature. So this can be considered as a pioneering work within this research area. The authors find identities in degree \(3, 4, 5\) and \(6\) which imply all the identities in degrees \(\leq 6\). They also prove the existence of further new identities in degree \(7\).Concerning the methodology of the work under review, this is rather multidisciplinary. The representation theory of the symmetric group plays a fundamental role. On the other hand, tools as the Hermite normal form of an integer matrix, the LLL algorithm for lattice reduction and the Chinese remainder theorem are also essential for this research. Computer algebra is the final ingredient combining all of these techniques. Reviewer: Candido Martín González (Málaga) Cited in 8 Documents MSC: 17C70 Super structures 17B01 Identities, free Lie (super)algebras 17-08 Computational methods for problems pertaining to nonassociative rings and algebras Keywords:Lie superalgebra; Jordan superalgebra; identities; LLL algorithm; symmetric group representation; Hermite normal form; Chinese remainder theorem Citations:Zbl 0860.17007 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Baranov, A. A., Volichenko algebras and nonhomogeneous subalgebras of Lie superalgebras, Siberian Math. J., 36, 5, 859-868 (1995) · Zbl 0860.17007 [2] Clifton, J. M., A simplification of the computation of the natural representation of the symmetric group \(S_n\), Proc. Amer. Math. Soc., 83, 2, 248-250 (1981) · Zbl 0443.20013 [3] Cohen, H., A Course in Computational Algebraic Number Theory, Grad. Texts in Math., vol. 138 (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0786.11071 [4] Leites, D.; Serganova, V., Metasymmetry and Volichenko algebras, Phys. Lett. B, 252, 1, 91-96 (1990) · Zbl 0753.17006 [5] Leites, D.; Serganova, V., Symmetries wider than supersymmetry, (Noncommutative Structures in Mathematics and Physics. Noncommutative Structures in Mathematics and Physics, Kiev, 2000. Noncommutative Structures in Mathematics and Physics. Noncommutative Structures in Mathematics and Physics, Kiev, 2000, NATO Sci. Ser. II Math. Phys. Chem., vol. 22 (2001), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 13-30 · Zbl 1248.17015 [6] Lenstra, A. K.; Lenstra, H. W.; Lovász, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 4, 515-534 (1982) · Zbl 0488.12001 [7] Serganova, V., Simple Volichenko algebras, (Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989. Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989, Contemp. Math., vol. 131 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 155-160 · Zbl 0765.17007 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.