Non-commutative Courant algebroids and Quiver algebras

TL;DR

This paper develops a non-commutative geometric framework for Courant algebroids using differential-graded symplectic structures, extending classical constructions to the non-commutative setting with applications to graded quivers.

Contribution

It introduces a non-commutative analog of Courant algebroids via a differential-graded symplectic approach, connecting double derivations and non-commutative forms.

Findings

Constructed non-commutative Courant algebroids using graded quivers.

Expressed BRST charge in terms of double Poisson algebras.

Provided examples of exact non-commutative Courant algebroids.

Abstract

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 \v{S}evera 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.