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.