Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

In this paper, first we show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between adjoint representation and coadjoint representation. Then introduce notions of relative Rota-Baxter operators on algebras pre-Lie-Yamaguti algebras. We prove gives rise to naturally operator algebra. Finally, study symplectic structures algebra, which give as well As applications, phase spaces algebras, there is one-to-one correspondence Manin triples