Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang–Baxterequations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces. Among these, particularly well behaved ones have deformation parameter u in a two-dimensional here S2. Quotients include seven spheres as well as noncommutative quaternionic tori. There is invariance for an action of SU(2) × SU(2) on the torus in parallel with the action of U(1) × U(1) on a ‘complex’ noncom-mutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case