Shuangjian Guo

We establish a cohomology theory for Reynolds Lie triple systems by first developing their representation theory and then assembling the corresponding cochains into a suitable complex. This construction is subsequently applied to deformation and extension problems. In particular, infinitesimal formal deformations are described by degree-three cocycles, while equivalence classes of abelian extensions are determined by the associated cohomology classes. We further introduce Reynolds Lie triple 2-systems and show that the skeletal case is governed by degree-five cocycles of Reynolds Lie triple systems.

In this paper, we introduce the notion of modified Rota–Baxter operators of nonzero weight on [ ]-Lie algebras and provide some examples. Next, we give various constructions of modified Rota–Baxter operators of nonzero weight according to constructions of [ ]-Lie algebras. Furthermore, we define a cohomology of modified Rota–Baxter operators of nonzero weight on [ ]-Lie algebras with coefficients in a suitable representation. As an application, we study formal deformations of modified Rota–Baxter operators of nonzero weight that are generated by the above-defined cohomology. In the final part of the paper, we construct two [ ]-algebra structures whose Maurer–Cartan elements correspond to relative and absolute modified Rota–Baxter [ ]-Lie algebra structures of nonzero weight, respectively. Lastly, we compare our [ ]-algebraic approach with the deformation-controlling [ ]-algebra for relative Rota–Baxter [ ]-Lie operators developed by Hou, Sheng, and Zhou.

In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. Conversely, a 3-tri-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the quotient map is an embedding tensor. Furthermore, any 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. Next, we introduce the notion of 3-tri-Leibniz dialgebras and demonstrate that homomorphic embedding tensors inherently induce 3-tri-Leibniz dialgebras. Finally, we study the linear deformations of embedding tensors by defining first cohomology.

The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter LieDer and weighted Rota-Baxter AssDer pairs. Then we construct cohomologies of weighted Rota-Baxter LieDer pairs, weighted Rota-Baxter AssDer pairs and we discuss the relation between their cohmologies. Finally, as an application, we study deformations of both of them.

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

In this paper, first we introduce the concept of modified Rota-Baxter Lie-Yamaguti algebras. Then the cohomology of a modified Rota-Baxter Lie-Yamaguti algebra with coefficients in a suitable representation is established. As applications, the formal deformations and abelian extensions of modified Rota-Baxter Lie-Yamaguti algebras are studied using the second cohomology group.