Twisted Rota-Baxter families on Lie-Yamaguti algebras and NS-Lie-Yamaguti family algebras
Wen Teng
TL;DR
The paper develops a framework for Γ-twisted Rota-Baxter families on Lie-Yamaguti algebras indexed by a commutative semigroup $Ω$, and shows how NS-Lie-Yamaguti family algebras arise as their underlying structures through explicit representation-theoretic data. It then introduces Ω-Lie-Yamaguti algebras and a corresponding cohomology theory, providing a robust toolset for studying representations and deformations of these family algebras. A major contribution is a graph-based and tensor-product treatment that connects twisted Rota-Baxter families to NS-Lie-Yamaguti family algebras, together with a comprehensive deformation theory controlled by the newly developed cohomology, including criteria for rigidity. Overall, the work unifies family-version algebraic structures with cohomological methods to analyze deformations and representations in the Lie-Yamaguti setting.
Abstract
In this paper, we first introduce twisted Rota-Baxter families on Lie-Yamaguti algebras indexed by a commutative semigroup $Ω$. Then, we study NS-Lie-Yamaguti family algebras as the underlying structures of twisted Rota-Baxter families. Finally, we investigate the cohomology of a twisted Rota-Baxter family. This cohomology can also be seen as the cohomology of a certain $Ω$-Lie-Yamaguti algebras with coefficients in an appropriate representation. As applications, we consider the deformations of twisted Rota-Baxter families from the cohomological points of view.