In this paper, we introduce the concept and representation of modified $ \lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ \lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ \lambda $-differential Lie triple systems.

In this paper, first we introduce the notion of embedding tensors on Lie triple systems and show that embedding tensors induce naturally 3-Leibniz algebras. Next, we construct a Lie 3-algebra whose Maurer-Cartan elements are embedding tensors. Then, we have the twisted? L-algebra that governs deformations of embedding tensors. Following this, we establish the cohomology of an embedding tensor on a Lie triple system and realize it as the cohomology of the induced 3-Leibniz algebra with coefficients in a suitable representation. As applications, we consider infinitesimal and finite order deformations of an embedding tensor on a Lie triple system from a cohomological viewpoint.

In this paper, the notion of generalized Reynolds operators on Lie-Yamaguti algebras is introduced, and the cohomology of a generalized Reynolds operator is established. The formal deformations of a generalized Reynolds operator are studied using the first cohomology group. Then, we show that a Nijenhuis operator on a Lie-Yamaguti algebra gives rise to a representation of the deformed Lie-Yamaguti algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed Lie-Yamaguti algebra. We also introduce the notion of a Reynolds operator on a Lie-Yamaguti algebra, which can serve as a special case of generalized Reynolds operators on Lie-Yamaguti algebras.

In this paper, we propose the representation and cohomology of modified $\lambda$-differential 3-Lie algebras. As their applications, the linear deformations, abelian extensions and $T^{\ast }$-extensions of modified $\lambda$-differential 3-Lie algebras are also studied.

The purpose of the present paper is to investigate cohomology of modified λ-differential Jacobi–Jordan algebras. First, we introduce the concept and representations of modified λ-differential Jacobi–Jordan algebras. Moreover, we define a lower order cohomology theory for modified λ-differential Jacobi–Jordan algebras. As applications of the proposed cohomology theory, formal deformations of modified λ-differential Jacobi–Jordan algebras are obtained and the rigidity of a modified λ-differential Jacobi–Jordan algebra is characterized by the vanishing of the second cohomology group. Also, abelian extensions of modified λ-differential Jacobi–Jordan algebras are classified by second-order cohomology. Furthermore, we study T*-extensions of modified λ-differential Jacobi–Jordan algebras.