Twisted exterior derivatives for universal enveloping algebras I

TL;DR

This paper develops a twisted exterior calculus for universal enveloping algebras using a representation of a Lie algebra, extending classical differential calculus to a noncommutative setting with deformed Leibniz rule.

Contribution

It introduces a canonical extension of Lie algebra representations to a tensor product algebra and constructs a twisted exterior derivative with deformed Leibniz rule.

Findings

The exterior derivative squares to zero, preserving cohomological properties.

Commutators between differentials and coordinates are formal power series in derivatives.

Provides a framework for noncommutative differential calculus in Lie algebra contexts.

Abstract

Consider any representation $\phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $\Lambda(g)$. We show that the representation $\phi$ extends canonically to the representation $\tilde\phi$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the Leibniz rule is deformed.