Associativity constraints, braidings and quantizations of modules with grading and action

(Lobachevskii Journal of Mathematics, Volume XXIII)

We study quantizations, associativity constraints and braidings in the monoidal category of monoid graded modules over a commutative ring. All of them can be described in terms of the cohomology of the underlying (finite) monoid. The Fourier transform of finite groups gives a corresponding description in the monoidal category of modules with action by a group.