Given a commutative ring and an associative -algebra over , the tensor product is equipped with two bimodule structures, “outer” and “inner”. For the outer structure and for the inner . The two bimodule structures mutually commute. A -linear map is called a double derivation if it is also a map of -bimodules with respect to the outer bimodule structure (); thus the -module of all double derivations becomes an -bimodule with respect to the inner -bimodule structure.
The tensor algebra of the -bimodule (which is the free monoid on in the monoidal category of -bimodules) is a step in the definition of the deformed preprojective algebra?s of Bill Crawley-Boevey?. A theorem of van den Bergh says that for any associative the tensor algebra has a canonical double Poisson bracket?.