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

Abstract

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.