Modular vector fields in non-commutative geometry

TL;DR

This paper develops a non-commutative analogue of the modular vector field on Poisson manifolds, introducing a triple divergence map and applying it to describe a groupoid version of Turaev's loop operation.

Contribution

It introduces a novel non-commutative construction of modular vector fields using a triple divergence map derived from connections, extending Poisson geometry concepts.

Findings

Constructed a non-commutative modular vector field analogue.

Defined a triple divergence map from a connection on a linear category.

Provided an algebraic description of a groupoid version of Turaev's loop operation.

Abstract

We construct a non-commutative analogue of the modular vector field on a Poisson manifold for a given pair of a double bracket and a connection on a space of 1-forms. The key ingredient, the triple divergence map, is directly constructed from a connection on a linear category to deal with multiple base points. As an application, we give an algebraic description of the framed, groupoid version of Turaev's loop operation $\mu$ similar to the one obtained by Alekseev-Kawazumi-Kuno-Naef and the author.