Weyl n-algebras and the Kontsevich integral of the unknot

TL;DR

This paper explores the deformation of the Hochschild complex related to Lie algebras with scalar products, connects it to the Kontsevich integral of the unknot, and provides a new proof of the Duflo isomorphism.

Contribution

It explicitly calculates the first order deformation of the Hochschild complex differential and links it to knot invariants and the Duflo isomorphism.

Findings

The first order deformation contains the Duflo character.

The Wilson loop invariant of the unknot coincides with the Kontsevich integral.

Provides a new proof of the Duflo isomorphism.

Abstract

Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a $dg$-scheme, which is the spectrum of the Chevalley--Eilenberg algebra. In the first section we explicitly calculate the first order deformation of the differential on the Hochschild complex of the Chevalley--Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is hopefully equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.