The Kontsevich-Rosenberg principle for bi-symplectic forms

TL;DR

This paper discusses how bi-symplectic forms on associative algebras relate to geometric structures on their representation spaces, using the Van den Bergh functor to provide an alternative proof of their noncommutative symplectic nature.

Contribution

It offers an alternative proof that bi-symplectic forms satisfy the Kontsevich-Rosenberg principle via the Van den Bergh functor.

Findings

Bi-symplectic forms induce geometric structures on representation spaces.

Bi-symplectic algebras can be viewed as noncommutative symplectic manifolds.

The Van den Bergh functor formalizes the Kontsevich-Rosenberg principle.

Abstract

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.