\((A,\delta )\)-modules, Hochschild homology and higher derivations

In this paper, we develop the theory of modules over \((A,\delta )\), where A is an algebra and \(\delta :A\longrightarrow A\) is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies \(HH_\bullet (A)\) of A into a module over \((A,\delta )\). We also consider modules over \((A,\Delta )\), where \(\Delta =\{\Delta ^n\}_{n\ge 0}\) is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A.