# Twisted exterior derivatives for universal enveloping algebras I

Zoran \v{S}koda

Arxiv ID: 0806.0978•Last updated: 8/18/2020

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.

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