Deformation of Koszul algebras and the Duflo Isomorphism theorem

Abstract

Let $\mathfrak g$ be a finite dimensional Lie algebra over a field $\mathbf k$, $U\mathfrak g$ be its enveloping algebra and $S\mathfrak g$ be the symmetric algebra on $\mathfrak g$. Extending the work of Braverman and Gaitsgory on the deformation of Koszul algebras and the Poincar{\'e}-Birkhoff-Witt theorem we obtain a generalized Duflo isomorphism which is valid also over fields of finite characteristic: $H_{\text{Lie}}^n(\mathfrak g, S\mathfrak g) \cong H_{\text{Hoch}}^n(U\mathfrak g,U\mathfrak g)$ for all $n < \operatorname{char}\mathbf k$. This implies, in particular, that Duflo's classic theorem, which is the special case in characteristic zero of dimension zero, in fact holds in all characteristics and the generalized theorem holds whenever $\dim \mathfrak g < \operatorname{char} \mathbf k$.