The twisting procedure

Abstract

This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, we settle the integration theory of complete pre-Lie algebras in order to describe this twisting procedure in terms of gauge group action. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of (homotopy) algebras. We also give a new presentation of the twisting procedure for operads à la Willwacher and we perform new homology computations of graph complexes.

arXiv commentNew format (short monography), 93 pages, minor corrections, submitted version

References in corpus (4)

• Rooted trees and an exponential-like series
• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• Deformation theory with homotopy algebra structures on tensor products
• Curved A-infinity-categories: adjunction and homotopy

Cited by in corpus (5)

• Deformation theory of Cohomological Field Theories
• The Homotopy Class of twisted $L_{\infty }$-morphisms
• Homotopy theory of curved operads and curved algebras
• Sheaves of E-infinity algebras and applications to algebraic varieties and singular spaces
• Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)