Jordan quadruple systems
Abstract
We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad {a,b,c,d} = abcd + dcba 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] = abcd - dcba. 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 Grobner bases, and Wedderburn decompositions of associative algebras.
Keywords
Cite
@article{arxiv.1402.5152,
title = {Jordan quadruple systems},
author = {Murray Bremner and Sara Madariaga},
journal= {arXiv preprint arXiv:1402.5152},
year = {2025}
}
Comments
30 pages; new material added on nonlinear identities for the anti-terrad (see pages 22-23 of the revised version)