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

Abstract

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

Relative Rota-Baxter operators of nonzero weights on Lie Triple Systems

“…In [17], the author showed the concept of Ooperators (relative Rota-Baxter operator) on Lie algebras. Then O-operators have been used to define cohomology on 3-Lie algebras, Lie triple system and Lie-Yamaguti algebras, see [6,21,23].…”