A family of algebraic operations extending the Turaev cobracket

TL;DR

This paper introduces a family of algebraic operations extending the Turaev cobracket, connecting non-commutative geometry, ribbon graphs, and Lie algebra cohomology, with applications to surface groups and free associative algebras.

Contribution

It develops a new family of algebraic maps parametrized by ribbon graphs, generalizing the Turaev cobracket and linking it to Lie algebra cohomology and non-commutative geometry.

Findings

Generalizes Turaev cobracket via ribbon graph parametrization

Identifies algebraic maps with standard generators of Lie algebra cohomology

Connects non-commutative geometric structures to surface group invariants

Abstract

We introduce a family of maps parametrised by certain ribbon graphs. It is based on a connection in non-commutative geometry and contains the double divergence as a special case. Applying the construction to the case of the group algebra of the fundamental group of a compact connected oriented surface with boundary, we obtain an algebraic generalisation of the Turaev cobracket. If the connection is flat, they define classes in the Lie algebra cohomology of the space of derivations. In the case of the free associative algebra, we show that they are canonically identified with the standard generators of the cohomology ring of the matrix Lie algebra $\mathfrak{gl}_n$.