On a homotopy version of the Duflo isomorphism

Abstract

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras. Moreover, it induces an isomorphism of algebras on the level of Lie algebra cohomology $H(\mathfrak{g},S\mathfrak{g})\rightarrow H(\mathfrak{g}, U\mathfrak{g})$. However, as shown by J. Alm and S. Merkulov, it cannot be extended in a universal way to an $A_\infty$-isomorphism between the corresponding Chevalley-Eilenberg complexes. In this paper, we give an elementary and self-contained proof of this fact using a version of M. Kontsevich's graph complex.