Deformation cohomology for Yetter-Drinfel’d modules and Hopf (bi)modules

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present name is taken from [10]). If A is a finite dimensional Hopf algebra then the category of left-right Yetter-Drinfel’d modules is equivalent to the category of left modules over D(A), the Drinfel’d double of A (see [6], [9]), even as braided tensor categories, and also to the category of Hopf bimodules over A (see [1], [11], [12], [17]). An important class of examples occurs as follows: if M is a finite dimensional vector space and R ∈ End(M ⊗ M) is a solution to the quantum Yang-Baxter equation, then the so-called “FRT construction” (see for instance [2]) associates to R a certain bialgebra A(R), and M becomes a left-right Yetter-Drinfel’d module over A(R) (see [5], [9]). In this paper we introduce a cohomology theory for left-right Yetter-Drinfel’d modules. If A is a bialgebra and M,N are left-right Yetter-Drinfel’d modules over A, we construct a double complex {Y (M,N)} whose total cohomology is the desired cohomology H(M,N). For M = N = k, this cohomology