Deformation maps of Quasi-twilled associative algebras

In this paper, we introduce two types of deformation maps of quasi-twilled associative algebras. Each type of deformation maps unify various operators on associative algebras. Right deformation maps unify modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms and crossed homomorphisms. Left deformation maps unify relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of associative algebras. Furthermore, we give the controlling algebra and the cohomology of these two types of deformation maps. On the one hand, we obtain some existing results for modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms, crossed homomorphisms, relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators and Reynolds operators. On the other hand, we also obtain some new results, such as the controlling algebra of a modified Rota-Baxter operator of weight $\lambda$ on an associative algebra, the controlling algebra and the cohomology of a deformation map of a matched pair of associative algebras.