Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces R N 1 × R R N 2 . 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 R 4 × R R 4 . Among these, particularly well behaved ones have deformation parameter u ∈ S 2 . Quotients include seven spheres S u 7 as well as noncommutative quaternionic tori T u H = S 3 × u S 3 . There is invariance for an action of SU ( 2 ) × SU ( 2 ) on the torus T u H in parallel with the action of U ( 1 ) × U ( 1 ) on a ‘complex’ noncommutative torus T θ 2 which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.