Jordan triple disystems
We take an algorithmic and computational approach to a basic problem
in abstract algebra: determining the correct generalization to dialgebras of a
given variety of nonassociative algebras. We give a simplified statement of
the KP algorithm introduced by Kolesnikov and Pozhidaev for extending
polynomial identities for algebras to corresponding identities for dialgebras.
We apply the KP algorithm to the defining identities for Jordan triple systems
to obtain a new variety of nonassociative triple systems, called Jordan triple
disystems. We give a generalized statement of the BSO algorithm introduced by
Bremner and Sanchez-Ortega for extending multilinear operations in an
associative algebra to corresponding operations in an associative dialgebra.
We apply the BSO algorithm to the Jordan triple product and use computer
algebra to verify that the polynomial identities satisfied by the resulting
operations coincide with the results of the KP algorithm; this provides a large
class of examples of Jordan triple disystems. We formulate a general
conjecture expressed by a commutative diagram relating the output of the KP and
BSO algorithms. We conclude by generalizing the Jordan triple product in a
Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to
verify that resulting structures provide further examples of Jordan triple
disystems. For this last result, we also provide an independent theoretical
proof using Jordan structure theory.
M. R. Bremner < bremner@math.usask.ca >
R. Felipe < raulf@cimat.mx >
J. Sánchez-Ortega < jsanchez@agt.cie.uma.es >