Noncommutative Euclidean spaces

TL;DR

This paper introduces a framework for noncommutative Euclidean spaces, explores their products, and constructs noncommutative spheres, quaternionic planes, and tori, extending classical geometric concepts into the noncommutative setting.

Contribution

It provides a complete characterization of noncommutative products of Euclidean spaces and constructs new noncommutative geometric objects like spheres and quaternionic manifolds.

Findings

Noncommutative Euclidean spaces are well-defined and characterized.

Conditions for noncommutative products of Euclidean spaces are fully solved.

Noncommutative spheres and quaternionic manifolds are constructed as noncommutative spherical manifolds.

Abstract

We give a definition of noncommutative finite-dimensional Euclidean spaces $\mathbb R^n$. We then remind our definition of noncommutative products of Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ which produces noncommutative Euclidean spaces $\mathbb R^{N_1+N_2}$. We solve completely the conditions defining the noncommutative products of the Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ and prove that the corresponding noncommutative unit spheres $S^{N_1+N_2-1}$ are noncommutative spherical manifolds. We then apply these concepts to define "noncommutative" quaternionic planes and noncommutative quaternionic tori on which acts the classical quaternionic torus $T^2_{\mathbb H}=U_1(\mathbb H)\times U_1(\mathbb H)$