Affine noncommutative geometry

TL;DR

This paper introduces noncommutative geometry from an affine perspective, focusing on free analogues of spheres and their algebraic submanifolds, and explores various related geometries including easy quantum geometries.

Contribution

It provides an affine coordinate-based introduction to noncommutative geometry, emphasizing free analogues of spheres and their submanifolds, and discusses the interplay of classical, free, real, and complex geometries.

Findings

Identification of free analogues of spheres and submanifolds

Analysis of Haar integration functionals on these spaces

Discussion of the relationships among real, complex, classical, and free geometries

Abstract

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$ do have free analogues $S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$. There are many examples of real algebraic submanifolds $X\subset S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $\int:C(X)\to\mathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.