nLab
double derivation

Given a commutative ring kk and an associative kk-algebra AA over kk, the tensor product A kAA\otimes_k A is equipped with two bimodule structures, “outer” and “inner”. For the outer structure a o(bc) od=abcda\cdot_o(b\otimes c)\cdot_o d = a b\otimes c d and for the inner a i(bc) id=bdaca\cdot_i(b\otimes c)\cdot_i d = b d\otimes a c. The two bimodule structures mutually commute. A kk-linear map αHom k(A,AA)\alpha\in Hom_k(A,A\otimes A) is called a double derivation if it is also a map of AA-bimodules with respect to the outer bimodule structure (αAModA( AA A, AA kA A)\alpha\in A Mod A({}_A A_A,{}_A A\otimes_k A_A)); thus the kk-module Der(A,AA)Der(A,A\otimes A) of all double derivations becomes an AA-bimodule with respect to the inner AA-bimodule structure.

The tensor algebra T ADer(A,AA)T_A Der(A,A\otimes A) of the AA-bimodule Der(A,AA)Der(A,A\otimes A) (which is the free monoid on Der(A,AA)Der(A,A\otimes A) in the monoidal category of AA-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 AA the tensor algebra T ADer(A,AA)T_A Der(A,A\otimes A) 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 (71.104.230.172)