From the Lie operad to the Grothendieck-Teichmüller group Article Swipe

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Mathematics Homotopy Morphism Pure mathematics Functor Graph Group (periodic table) Lie group Rigidity (electromagnetism) Combinatorics Physics Quantum mechanics

Vincent Wolff ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2302.01787 · OA: W4319323674

We study the deformation complex of the standard morphism from the degree $d$ shifted Lie operad to its polydifferential version, and prove that it is quasi-isomorphic to the Kontsevich graph complex $\mathbf{GC}_d$. In particular, we show that in the case $d=2$ the Grothendieck-Teichmüller group $\mathbf{GRT_1}$ is a symmetry group (up to homotopy) of the aforementioned morphism. We also prove that in the case $d=1$ corresponding to the usual Lie algebras the standard morphism admits a unique homotopy non-trivial deformation which is described explicitly with the help of the universal enveloping construction. Finally we prove the rigidity of the strongly homotopy version of the universal enveloping functor in the Lie theory.