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TitleJordan quadruple systems
Author(s) Murray R. Bremner, Sara Madariaga
TypeArticle in Journal
AbstractAbstract We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad a , b , c , d = a b c d + d c b a as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are not consequences of the defining identities. We introduce four infinite families of finite dimensional Jordan quadruple systems, and construct the universal associative envelope for a small system in each family. We obtain analogous results for the anti-tetrad [ a , b , c , d ] = a b c d − d c b a . Our methods rely on computer algebra, especially linear algebra on large matrices, the LLL algorithm for lattice basis reduction, representation theory of the symmetric group, noncommutative Gröbner bases, and Wedderburn decompositions of associative algebras.
KeywordsJordan tetrad, Polynomial identities, Gröbner bases, Universal associative envelopes, Wedderburn decompositions
URL http://www.sciencedirect.com/science/article/pii/S0021869314002403
JournalJournal of Algebra
Pages51 - 86
Translation No
Refereed No