nLab double derivation

Overview

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 algebras of Bill Crawley-Boevey. A theorem of Michel 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.

Literature

category: algebra

Last revised on October 15, 2023 at 16:27:46. See the history of this page for a list of all contributions to it.