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A family of algebraic operations extending the Turaev cobracket

Toyo Taniguchi

TL;DR

The paper constructs a family of higher divergence maps $\mathrm{Div}^{\nabla}_k$ and their pullbacks $\delta^{\psi,\nabla}_k$ to extend the Turaev cobracket in a non-commutative, algebraic framework. It connects these operators to ribbon-graph calculus, showing that for tensor algebras with a skew pairing the higher-divergence operations coincide with the $L_k$ ribbon-graph operators, and that flat connections yield Lie algebra cocycles with even $k$ vanishing; it also proves nontrivial Chevalley–Eilenberg cohomology classes for odd $k$ in the tensor algebra case. The paper develops non-commutative de Rham theory with endomorphism-valued forms, linking the divergences to trace maps and CE cohomology, and establishes a bridge between algebraic loop operations, ribbon graphs, and derivation cohomology. Altogether, it provides an algebraic generalization of the Turaev cobracket and a framework marrying non-commutative geometry with infinitesimal actions on free associative algebras.

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 $\mathfrak{gl}_n$.

A family of algebraic operations extending the Turaev cobracket

TL;DR

The paper constructs a family of higher divergence maps and their pullbacks to extend the Turaev cobracket in a non-commutative, algebraic framework. It connects these operators to ribbon-graph calculus, showing that for tensor algebras with a skew pairing the higher-divergence operations coincide with the ribbon-graph operators, and that flat connections yield Lie algebra cocycles with even vanishing; it also proves nontrivial Chevalley–Eilenberg cohomology classes for odd in the tensor algebra case. The paper develops non-commutative de Rham theory with endomorphism-valued forms, linking the divergences to trace maps and CE cohomology, and establishes a bridge between algebraic loop operations, ribbon graphs, and derivation cohomology. Altogether, it provides an algebraic generalization of the Turaev cobracket and a framework marrying non-commutative geometry with infinitesimal actions on free associative algebras.

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 .

Paper Structure

This paper contains 5 sections, 18 theorems, 49 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Divergence Maps and a Family of Operations
  3. Props of Ribbon Graphs
  4. Non-commutative de Rham Cohomology and Vector Bundles
  5. The Infinitesimal Action and Divergences

Key Result

Theorem 1

Let $\nabla\!_W$ be the canonical flat connection given in Definition def:connW on the space of non-commutative $1$-forms on $T(W)$. Then, the map coincides with the operation associated with the unique bivalent connected ribbon graph with $k$ vertices and $k$ edges.

Figures (1)

  • Figure 1: The ribbon graph $L_k\in\mathcal{R}^k_{2,k}$. The labelling of the edges is suppressed, as any choice of these gives the same element in $\mathcal{RG}ra_1(2,k)$.

Theorems & Definitions (41)

  • Theorem : Theorem \ref{['thm:ribbonop']}
  • Theorem : Theorem \ref{['thm:kcocyc']}
  • Theorem : Theorem \ref{['thm:dercohom']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 31 more