Deformation of Koszul algebras and the Duflo Isomorphism theorem

TL;DR

This paper extends the Duflo isomorphism to fields of finite characteristic, showing it holds generally for Lie algebras with dimension less than the characteristic, thus broadening its applicability beyond characteristic zero.

Contribution

The authors generalize the Duflo isomorphism to finite characteristic fields, connecting Lie algebra cohomology with Hochschild cohomology in a new setting.

Findings

Duflo's theorem holds in all characteristics for finite-dimensional Lie algebras.

The generalized isomorphism applies when the algebra's dimension is less than the field's characteristic.

The work extends deformation theory of Koszul algebras to new algebraic contexts.

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 $\dim \mathfrak g < \operatorname{char} \mathbf k$.