nLab double derivation

Overview

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 $a\cdot_o(b\otimes c)\cdot_o d = a b\otimes c d$ and for the inner $a\cdot_i(b\otimes c)\cdot_i d = b d\otimes a c$. The two bimodule structures mutually commute. A $k$-linear map $\alpha\in Hom_k(A,A\otimes A)$ is called a double derivation if it is also a map of $A$-bimodules with respect to the outer bimodule structure ($\alpha\in A Mod A({}_A A_A,{}_A A\otimes_k A_A)$); thus the $k$-module $Der(A,A\otimes A)$ of all double derivations becomes an $A$-bimodule with respect to the inner $A$-bimodule structure.

The tensor algebra $T_A Der(A,A\otimes A)$ of the $A$-bimodule $Der(A,A\otimes A)$ (which is the free monoid on $Der(A,A\otimes A)$ in the monoidal category of $A$-bimodules) is a step in the definition of the deformed preprojective algebras of Bill Crawley-Boevey. A theorem of Michel Van den Bergh says that for any associative $A$ the tensor algebra $T_A Der(A,A\otimes A)$ has a canonical double Poisson bracket.

Literature

• 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
• William Crawley-Boevey, Pavel Etingof, Victor Ginzburg, Noncommutative geometry and quiver algebras, Adv. Math. 209:1 (2007) 274-336 doi
• Victor Ginzburg, Travis Schedler, Differential operators and BV structures in noncommutative geometry, Sel. Math. New Ser. 16, 673–730 (2010) doi
• Victor Ginzburg, Travis Schedler, Boris Tsygan, Free products, cyclic homology, and the Gauss–Manin connection, Advances in Mathematics 231:3-4 (2012) 2352–2389 doi

category: algebra