Jordan quadruple systems
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 Groebner bases, and
Wedderburn decompositions of associative algebras.
Murray Bremner < bremner@math.usask.ca >
Sara Madariaga < madariaga@math.usask.ca >