Title: Noncommutative products of Euclidean spaces,
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.