nLab
double derivation

Given a commutative ring $k$ and an associative $k$-algebra $A$ over $k$, the tensor product $A{\otimes }_{k}A$ is equipped with two bimodule structures, “outer” and “inner”. For the outer structure $a{\cdot }_o(b\otimes c){\cdot }_{o}d=ab\otimes cd$ and for the inner $a{\cdot }_i(b\otimes c){\cdot }_{i}d=bd\otimes ac$. The two bimodule structures mutually commute. A $k$-linear map $\alpha \in {\mathrm{Hom}}_k(A,A\otimes A)$ is called a double derivation if it is also a map of $A$-bimodules with respect to the outer bimodule structure ($\alpha \in A\mathrm{Mod}A({}_A{A}_A,{}_{A}A{\otimes }_k{A}_A)$); thus the $k$-module $\mathrm{Der}(A,A\otimes A)$ of all double derivations becomes an $A$-bimodule with respect to the inner $A$-bimodule structure.

The tensor algebra $T_A\mathrm{Der}(A,A\otimes A)$ of the $A$-bimodule $\mathrm{Der}(A,A\otimes A)$ (which is the free monoid on $\mathrm{Der}(A,A\otimes A)$ 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 $T_A\mathrm{Der}(A,A\otimes A)$ 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