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Let $B$ be a commutative algebra and $A$ be a $B$-algebra (determined by an algebra homomorphism $\varepsilon:B\rightarrow A$). M. D. Staic introduced a Hochschild like cohomology $H^{\bullet}((A,B,\varepsilon);A)$ called secondary Hochschild cohomology, to describe the non-trivial $B$-algebra deformations of $A$. J. Laubacher et al later obtained a natural construction of a new chain (and cochain) complex $\overline{C}_{\bullet}(A,B,\varepsilon)$ (resp. $\overline{C}^{\bullet}(A,B,\varepsilon)$) in the process of introducing the secondary cyclic (co)homology. It turns out that unlike the classical case of associative algebras (over a field), there exist different (co)chain complexes for the $B$-algebra $A$. In this paper, we establish a connection between the two (co)homology theories for $B$-algebra $A$. We show that the pair $\big(H^{\bullet}((A,B,\varepsilon);A),HH_{\bullet}(A,B,\varepsilon)\big)$ forms a non-commutative differential calculus, where $HH_{\bullet}(A,B,\varepsilon)$ denotes the homology of the complex $\overline{C}_{\bullet}(A,B,\varepsilon)$., Comment: Any comments are welcome

We study formal deformations of hom-Lie-Rinehart algebras. The associated deformation cohomology that controls deformations is constructed using multiderivations of hom-Lie-Rinehart algebras., Comment: Any comments are welcome

In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $\alpha$- central extensions, and, lifting of automorphisms and $\alpha$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $\alpha^k$-derivations to the central extensions for hom-Lie algebras., Comment: 22 pages

We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and then define a hom-Poisson homology associated to a hom-Poisson manifold.

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism., Comment: Revised version, 23 pages

In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leibniz cohomology., Comment: 23 pages, any comments are welcome