Emergent version of Drinfeld's associator equations
Yusuke Kuno
TL;DR
This work introduces an emergent version of Drinfeld’s associator equations within a topological framework of emergent braids and subquotients of the Drinfeld–Kohno Lie algebra. It defines ${\sf grt}_1^{\mathrm{em}}$, the emergent Grothendieck–Teichmüller space, and constructs an emergent map ${\nu}^{\mathrm{em}}$ into the symmetric part of the Kashiwara–Vergne Lie algebra ${\sf krv}_2^{\rm sym}$, proving a graded linear isomorphism onto its image. The authors develop the emergent DK Lie algebra ${\mathsf{edk}}_{m,n}$ and its operadic structure, then lift Drinfeld’s associator data to homomorphic expansions for mixed braids ${\bf PaMB}$, culminating in linearized pentagon relations that govern the emergent setting. By connecting loop operations on punctured disks to KV theory, they reinterpret ${\sf krv}_n$ and ${\sf krv}_n^{0}$ in terms of the Goldman–Turaev framework and related derivations, following AKKN’s program. The main result thus situates emergent braids as a bridge between Drinfeld associators, KV theory, and expansions for tangles, with potential implications for broader relations among these theories and their topological incarnations.
Abstract
The works of Alekseev and Torossian [AT] and Alekseev, Enriquez, and Torossian [AET] show that any solution of Drinfeld's associator equations gives rise to a solution of the Kashiwara-Vergne equations in an explicit way. We introduce a weak version of Drinfeld's associator equations that we call the emergent version of the original equations. It is shown that solutions to the resulting linearized emergent Drinfeld's equations still lead to solutions to the linearized Kashiwara-Vergne equations. The emergent Drinfeld equations arise within a natural topological context of emergent braids, which we discuss. Our results are adjacent to the results of Bar-Natan, Dancso, Hogan, Liu and Scherich [BDHLS] on the relationship between emergent tangles and the Goldman-Turaev Lie bialgebra. We hope that in time our results will play a role in relating several bodies of work, on Drinfeld associators, Kashiwara-Vergne equations, and on expansions for classical tangles, for w-tangles, and for the Goldman-Turaev Lie bialgebra.