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

Abstract

In this paper, we develop the theory of modules over (A, δ), where A is an algebra and δ : A −→ A is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies H H • (A) of A into a module over (A, δ). We also consider modules over (A,), where = { n } n≥0 is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A.

Extensions, deformation and categorification of $\text{AssDer}$ pairs

“…We denote the corresponding cohomology groups by H n AssDer (A, M ), for n ≥ 1. It is known that the Hochschild cohomology of an associative algebra A with coefficients in itself carries a degree −1 graded Lie bracket (6). We prove a similar result for the cohomology of an AssDer pair using the bracket (6).…”

“…It is also known as a differential algebra in the literature and is itself a significant object of study in areas such as differential Galois theory [22]. Recently, the authors in [6] defined Hochschild homology for AssDer pairs motivated from noncommutative geometry.…”

“…We relate the cohomology of an AssDer pair with the cohomology of the corresponding commutator LieDer pair. This cohomology together with the homology defined in [6] might be starting point to study cyclic theory for AssDer pairs. Additionally, we define and study categorified derivation on a categorified associative algebra and relate them with homotopy derivations of 2-term A ∞ -algebras.…”