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A gentle introduction to Drinfel'd associators

Martin Bordemann, Andrea Rivezzi, Thomas Weigel

TL;DR

This work presents a self-contained, elementary treatment of Drinfeld's associator by develop­ing a formal parallel-transport framework associated with the Knizhnik-Zamolodchikov connection. The associator $\Phi(A,B)\in\mathcal{A}[[\lambda]]$ is defined as the non-singular part of a regularized parallel transport, with inversion symmetry $\Phi(A,B)^{-1}=\Phi(B,A)$ and the first nontrivial term appearing at $\lambda^2$. The Hexagon and Pentagon equations are proven by analyzing carefully constructed paths in the doubly punctured plane and a pentagon domain, respectively, and invoking flatness and limit arguments that cancel singular factors. The approach leverages formal linear ODEs, piecewise smooth analysis, and a robust limit framework to yield the associator identities without heavy differential-geometry prerequisites, highlighting connections to configuration spaces, infinitesimal braid relations, and KZ-type structures. The results underscore the role of the Drinfeld associator in quantization of Lie bialgebras and deformation theory, with ties to multiple zeta values and broader algebraic structures.

Abstract

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.

A gentle introduction to Drinfel'd associators

TL;DR

This work presents a self-contained, elementary treatment of Drinfeld's associator by develop­ing a formal parallel-transport framework associated with the Knizhnik-Zamolodchikov connection. The associator is defined as the non-singular part of a regularized parallel transport, with inversion symmetry and the first nontrivial term appearing at . The Hexagon and Pentagon equations are proven by analyzing carefully constructed paths in the doubly punctured plane and a pentagon domain, respectively, and invoking flatness and limit arguments that cancel singular factors. The approach leverages formal linear ODEs, piecewise smooth analysis, and a robust limit framework to yield the associator identities without heavy differential-geometry prerequisites, highlighting connections to configuration spaces, infinitesimal braid relations, and KZ-type structures. The results underscore the role of the Drinfeld associator in quantization of Lie bialgebras and deformation theory, with ties to multiple zeta values and broader algebraic structures.

Abstract

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.
Paper Structure (16 sections, 21 theorems, 188 equations, 2 figures)

This paper contains 16 sections, 21 theorems, 188 equations, 2 figures.

Table of Contents

  1. Some analysis for formal parallel transports
  2. Elementary analysis of piecewise $\mathcal{C}^\infty$-functions
  3. Formal linear ODE's
  4. Norms and limits
  5. Formal connections and parallel transports
  6. Flat formal connections
  7. The Drinfel'd associator and its identities
  8. The Knizhnik-Zamolodchikov connection and the Drinfel'd-Kohno (Lie) algebras
  9. The Drinfel'd associator: Definition and elementary properties
  10. The Hexagon Equation
  11. The Pentagon Equation
  12. Some Proofs of Theorems in Section 1.
  13. Proof of Proposition \ref{['PFormConnComplexPullBackEqualReal']}
  14. Proof of Proposition \ref{['PFlatConnPullBackOfFlatIsFlat']}
  15. Proof of Theorem \ref{['TFormFlatPathIndep']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $a,b \in \mathbb{R}$ with $a<b$, and let $D \subset \mathbb{R}$ be a finite set such that $\{a,b\}\subset D\subset [a,b]$. Then for any $f,g\in C^\infty_D([a,b],\mathbb{C})$ and $h\in C^{\infty}_D([a,b],\mathbb{C})^0$ the following holds:

Figures (2)

  • Figure 1: The paths (\ref{['EqHexagonSixPaths']}) at $\delta=1/4$
  • Figure 2: The paths (\ref{['EqPentEqAffinePathsOfPentagon']}) in the $x_2$-$x_3$-plane

Theorems & Definitions (41)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 31 more