Quasi-twilled associative algebras, deformation maps and their governing algebras

A quasi-twilled associative algebra is an associative algebra A whose underlying vector space has a decomposition A = A B such that B A is a subalgebra. In the first part of this paper, we give the Maurer-Cartan characterization and introduce the cohomology of a quasi-twilled associative algebra. In a quasi-twilled associative algebra A, a linear map D: A B is called a strong deformation map if Gr (D) A is a subalgebra. Such a map generalizes associative algebra homomorphisms, derivations, crossed homomorphisms and the associative analogue of modified r-matrices. We introduce the cohomology of a strong deformation map D unifying the cohomologies of all the operators mentioned above. We also define the governing algebra for the pair (A, D) to study simultaneous deformations of both A and D. On the other hand, a linear map r: B A is called a weak deformation map if Gr (r) A is a subalgebra. Such a map generalizes relative Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators, left-averaging operators and right-averaging operators. Here we define the cohomology and governing algebra of a weak deformation map r (that unify the cohomologies of all the operators mentioned above) and also for the pair (A, r) that govern simultaneous deformations.