Given a commutativering 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?.
Michel Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), 5711–5769, arXiv:math.AG/0410528
Anne Pichereau, Geert Van de Weyer, Double Poisson cohomology of path algebras of quivers, J. Alg. 319, 5 (2008), 2166–2208 (doi)
Jorge A. Guccione, Juan J. Guccione, A characterization of quiver algebras based on double derivations, arXiv:0807.1148
Revised on July 24, 2009 01:05:48
by Toby Bartels