Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

Abstract

In the present paper, we introduce the sheaf 𝔗𝔤 of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra 𝔤, which is a sheaf of non-commutative Fréchet algebras over the character space of 𝔤. We prove that 𝔗𝔤(D) is a localization over the universal enveloping algebra [Formula: see text] whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality.