Deformations of homotopy algebras

Let k be a field of characteristic zero, script O sign be a dg operad over k and let A be an script O sign-algebra. In this note we suggest a definition of a formal deformation functor of A Def_A : dgart^≤0(k) → Δ⁰Ens 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 script O sign 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.