Homotopical rigidity of the pre-Lie operad

Abstract

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no “non-obvious” degrees of freedom from either of the three points of view: deformations of maps to and from the “three graces of operad theory”, homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture.

Enriched pre-Lie operads and freeness theorems

“…Every operad PreLie C receives the unit map from the operad PreLie uCom * = PreLie, and therefore a map from the operad Lie; therefore, as pointed in [5], to each such operad one may apply the construction of operadic twisting [9]. It would be interesting to determine which of the operads PreLie C have interesting homotopical properties with respect to operadic twisting, for example, for which of them one has H 0 (Tw(PreLie C ), d Tw ) Lie, generalising the existing results for the pre-Lie and the brace operad, see [13].…”