The Kontsevich-Rosenberg principle for bi-symplectic forms

In this expository note, we explain the so-called Van den Bergh functor, which enables the formalization of the Kontsevich-Rosenberg principle, whereby a structure on an associative algebra has geometric meaning if it induces standard geometric structures on its representation spaces. Crawley-Boevey, Etingof and Ginzburg proved that bi-symplectic forms satisfy this principle; this implies that bi-symplectic algebras can be regarded as noncommutative symplectic manifolds. In this note, we use the Van den Bergh functor to give an alternative proof.