Operad and cohomology of associative algebras with generalized derivations

TL;DR

This paper introduces a new operad for associative algebras with generalized derivations, proves its Koszul property, and develops a cohomology theory that governs their deformations.

Contribution

It defines the operad encoding AsGDer triples, establishes its Koszul duality, and constructs the associated cohomology for deformation analysis.

Findings

The operad for AsGDer triples is Koszul.

A homotopy version of AsGDer triples is developed.

AsGDer cohomology controls formal deformations.

Abstract

An associative algebra with a generalized derivation is called an AsGDer triple. We introduce the operad that encodes AsGDer triples, and prove it is a Koszul operad. Using its Koszul dual cooperad, we introduce the homotopy version of AsGDer triples. As an application, we construct the AsGDer cohomology theory for AsGDer triples, and show that the formal deformation of an AsGDer triple is controlled by the AsGDer cohomology.