Unbounded operators, Lie algebras, and local representations

TL;DR

This paper investigates the conditions under which Lie algebras of unbounded operators in Hilbert spaces can be integrated into unitary group representations, extending previous results and introducing new invariants for Riemann surfaces.

Contribution

It extends existing integrability results for Lie algebras of unbounded operators and introduces new invariants for Riemann surfaces, even in the case of a single operator.

Findings

Extended integrability criteria for Lie algebras of unbounded operators

New invariants for certain Riemann surfaces

Results applicable to single operators as well

Abstract

We prove a number of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space. By integrability for a Lie algebra $\mathfrak{g}$, we mean that there is an associated unitary representation $\mathcal{U}$ of the corresponding simply connected Lie group such that $\mathfrak{g}$ is the differential of $\mathcal{U}$. Our results extend earlier integrability results in the literature; and are new even in the case of a single operator. Our applications include a new invariant for certain Riemann surfaces.