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

Abstract

In this paper, first we 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.

Keywords:

• Lie-Yamaguti algebra
• pre-Lie-Yamaguti algebra
• relative Rota-Baxter operator
• symplectic structure

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A99
• 17B38
• 17B60

Acknowledgments

This research was supported by NSFC (11922110).