Twisted exterior derivatives for universal enveloping algebras I

Abstract

Consider any representation $\varphi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^{\ast })$ of its dual. Consider the tensor product of $\hat{S}(g^{\ast })$ and the exterior algebra $\Lambda (g)$. We show that the representation $\varphi$ extends canonically to the representation $\tilde{\varphi }$ 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.

arXiv commentv2, v3: radical revisions, background more detailed, expositional corrections and, in the last section some mathematical. Slightly changed title, now part I with sequel planned. v4: 2.5 corrected/expanded to take care of completions, 15 pages. To appear in Contemporary Mathematics (2020), Proceedings of the conference "Representation Theory XVI", Dubrovnik IUC, June 19-25, 2019