# Noncommutative products of Euclidean spaces

## TL;DR

This paper introduces families of noncommutative Euclidean spaces constructed via quadratic algebras related to R-matrices satisfying Yang-Baxter equations, leading to well-behaved deformations including noncommutative spheres and tori.

## Contribution

It develops new coordinate algebras for noncommutative Euclidean spaces with properties like regularity and finite global dimension, including novel eight-dimensional examples and quaternionic structures.

## Key findings

- Eight-dimensional noncommutative Euclidean spaces parametrized by a sphere
- Construction of noncommutative seven-spheres and quaternionic tori
- Existence of solutions disjoint from classical cases

## Abstract

We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are deformations parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.