Sheaves of non-commutative smooth and holomorphic functions associated with the non-abelian two-dimensional Lie algebra

TL;DR

This paper constructs sheaves of noncommutative smooth and holomorphic functions on a specific space of representations for a solvable Lie algebra, extending previous work from nilpotent cases.

Contribution

It introduces sheaves of noncommutative functions for the non-abelian two-dimensional Lie algebra, expanding the framework to a non-nilpotent solvable Lie algebra.

Findings

Established sheaves of noncommutative functions on the representation space

Extended the sheaf construction from nilpotent to solvable Lie algebras

Provided explicit constructions for both holomorphic and smooth cases

Abstract

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fr\'echet--Arens--Michael algebras (of noncommutative holomorphic functions in the complex case and of noncommutative smooth functions in the real case). We construct similar sheaves (both versions, holomorphic and smooth) on a special space of representations for the Lie algebra of the group of affine transformations of the real line (which is the simplest nonnilpotent solvable Lie algebra).