Tamarkin’s construction is equivariant with respect to the action of the Grothendieck–Teichmueller group

Abstract

Recall that Tamarkin’s construction (Hinich, Forum Math 15(4):591–614, 2003, arXiv:math.QA/0003052 ; Tamarkin, 1998, arXiv:math/9803025 ) gives us a map from the set of Drinfeld associators to the set of homotopy classes of $$L_{\infty }$$ quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of Drinfeld (Algebra i Analiz 2(4):149–181, 1990) and Willwacher Invent Math 200(3):671–760, 2015 both the source and the target of this map are equipped with natural actions of the Grothendieck–Teichmueller group $${\mathsf {GRT}}_1$$ . In this paper, we use the result from Paljug (JHRS, 2015, arXiv:1305.4699 ) to prove that this map from the set of Drinfeld associators to the set of homotopy classes of $$L_{\infty }$$ quasi-isomorphisms for Hochschild cochains is $${\mathsf {GRT}}_1$$ -equivariant.