A New Approach to Jordan D-Bialgebras via Jordan-Manin Triples

Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated by the essential connection between Lie bialgebras and Manin triples,we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan-Manin triples called double constructions of pseudo-euclidean Jordan algebras.We also show that a Jordan D-bialgebra leads to the Jordan Yang-Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang-Baxter equation corresponds to an antisymmetric bilinear form,which we call a Jordan symplectic form on Jordan algebras.Furthermore,there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.