Cohomology and deformations of O-operators on Hom-associative algebras

Abstract

In this paper, we introduce the cohomology theory of O-operators on Homassociative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an O-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an O-operator is introduced to characterize trivial infinitesimal deformations.

Representations and relative Rota-Baxter operators of Hom-Leibniz Poisson algebras

“…Until the end of this section, we state and prove some results affirmed in [6]. These results will be used in the last section on this paper.…”

“…Similarly, we prove φλ r β (x) = λ r β (α(x))φ and hence, we obtain (5) for V β . Next, by (6) in V and the fact that β is a morphism, we get: (6) holds and similarly, (7) holds for V β . Finally, using αβ = βα and (8) for V, we compute…”

“…It is found that, those operators have many applications in Connes-Kreimers algebraic approach to the renormalization in perturbative quantum field theory [7]. A generalization of Rota-Baxter operators called relative Rota-Baxter operators (or O-operators) has been introduced for left Hom-Leibniz algebras [8] where the graded Lie algebra that characterizes those operators as Maurer-Cartan elements is constructed whereas in [6], the cohomology theory of O-operators on Hom-associative algebras are found. In this paper, from the representations of Hom-associative and Hom-Leiniz algebras, we will establish those of Hom-Leibniz-Poisson algebras.…”

Representations and formal deformations of Hom-Leibniz algebras

“…Let give another example of O-operators of Hom-Jacobi-Jordan algebras. As Hom-associative algebras case [9], let give some characterizations of O-operators on Hom-Jacobi-Jordan algebras. Proposition 3.22.…”

“…As Hom-associative algebras case [9], let give some characterizations of O-operators on left Hompre-Jacobi-Jordan algebras. The following result shows that an O-operator can be lifted up the Rota-Baxter operator.…”