Almost representations of algebras and quantization

TL;DR

This paper introduces the concept of almost representations for Lie algebras and quantum tori, demonstrating they are close to true representations, and applies this to show geometric quantizations are nearly conjugate in the semi-classical limit.

Contribution

It develops the theory of almost representations and proves their proximity to genuine representations, with applications to geometric quantization.

Findings

Almost representations are close to true irreducible representations.

Geometric quantizations of the sphere and torus are conjugate in the semi-classical limit.

Establishes an Ulam-stability type phenomenon for these algebraic structures.

Abstract

We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.