Non-commutative Courant algebroids and Quiver algebras

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In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commutative 2-forms of weight 1, and in terms of our non-commutative Courant algebroids for graded bi-symplectic non-commutative 2-forms of weight 2 (here, the grading, or ghost degree, is called weight). We then apply our formalism to obtain examples of exact non-commutative Courant algebroids, using appropriate graded quivers equipped with bi-symplectic forms of weight 2, with a possible twist by a closed Karoubi-de Rham non-commutative differential 3-form.