We introduce the notion of a BV-operator) is determined by ∆ in a manner similar to the BV-formalism. As an application, we produce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra A over a commutative algebra B . In this case, we also show that the operator ∆ • corresponds to Connes' operator.

We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base-the so called 'versal deformation'-which induces all other deformations of the given algebra.In memory of our good friend Jean-Louis Loday

O -operators (also known as relative Rota–Baxter operators) on Lie algebras have several applications in integrable systems and the classical Yang–Baxter equations. In this article, we study O-operators on hom-Lie algebras. We define a cochain complex for O-operators on hom-Lie algebras with respect to a representation. Any O-operator induces a hom-pre-Lie algebra structure. We express the cochain complex of an O-operator in terms of the specific hom-Lie algebra cochain complex. If the structure maps in a hom-Lie algebra and its representation are invertible, then we can extend the above cochain complex to a deformation complex for O-operators by adding the space of zero cochains. Subsequently, we study formal deformations of O-operators on regular hom-Lie algebras in terms of the deformation cohomology. In the end, we deduce deformations of s-Rota–Baxter operators (of weight 0) and skew-symmetric r-matrices on hom-Lie algebras as particular cases of O-operators on hom-Lie algebras.

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The purpose of this paper is to provide a cohomology of n-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.