2022/04/14 by Yunhe Sheng, Jia Zhao · 16 citations
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Lie conformal algebra #Mathematics #Pure mathematics #Symplectic geometry
paper · doi:10.1080/00927872.2022.2057517
published in Communications in Algebra 50(9), 4056-4073 (Taylor & Francis)
openalex publication_date 2022/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
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.