Deformations of Homotopy Algebras

Abstract

Let k be a field of characteristic zero, 𝒪 be a dg operad over k and let A be an 𝒪-algebra. In this note we suggest a definition of a formal deformation functor of A

from the category of artinian local dg algebras to the category of simplicial sets. This functor generalizes the classical deformation functor for an algebra over a linear operad. In the case 𝒪 and A are non-positively graded, we prove that Def _A is governed by the tangent Lie algebra T _A which can be calculated as the Lie algebra of derivations of a cofibrant resolution of A. An example shows that the result does not necessarily hold without the non-positivity condition.

Key Words:

• Formal deformations
• dg Lie algebra
• Operad algebras
• Simplicial categories

Acknowledgment

This work was made during my stay at the Max-Planck Institut für Mathematik at Bonn. I express my gratitude to the Institute for the hospitality and excellent working conditions.

Notes

^#Communicated by M. Cohen.