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

Revised on July 24, 2009 01:05:48
by Toby Bartels
(71.104.230.172)