Formality theorem on infinite-dimensional spaces and a conjecture of B.Feigin

Let An be the free associative algebra with n generators over C, consider the Lie algebra A1 of its outer derivations (the derivations modulo the inner derivations). Let A0 be its quasi-classical limit, that it the Lie algebra of outer derivations of the free Poisson algebra with n generators over C. Boris Feigin conjectured around 1998 that the Lie algebras A0 and A1 are isomorphic. It turns out that the Kontsevich’s formality theorem for a finite-dimensional vector space V follows from the Feigin’s conjecture, but not vise versa. In this paper we prove the Feigin’s conjecture. To prove it we construct Kontsevich formality map for any infinite-dimensional nonnegatively graded vector space with finite-dimensional grading components.