Noncommutative products of Euclidean spaces

Abstract

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces RN1×RRN2. 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 R4× RR4. Among these, particularly well behaved ones have deformation parameter u∈ S2. Quotients include seven spheres Su7 as well as noncommutative quaternionic tori TuH=S3×uS3. There is invariance for an action of SU (2) × SU (2) on the torus TuH in parallel with the action of U (1) × U (1) on a ‘complex’ noncommutative torus Tθ2 which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.