double derivation

Given a commutative ring k and an associative k-algebra A over k, the tensor product A kA is equipped with two bimodule structures, “outer” and “inner”. For the outer structure a o(bc) od=abcd and for the inner a i(bc) id=bdac. The two bimodule structures mutually commute. A k-linear map αHom k(A,AA) is called a double derivation if it is also a map of A-bimodules with respect to the outer bimodule structure (αAModA( AA A, AA kA A)); thus the k-module Der(A,AA) of all double derivations becomes an A-bimodule with respect to the inner A-bimodule structure.

The tensor algebra T ADer(A,AA) of the A-bimodule Der(A,AA) (which is the free monoid on Der(A,AA) in the monoidal category of A-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 A the tensor algebra T ADer(A,AA) 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 (