Solutions to the quantum Yang--Baxter equation and related deformations

Starting with the classical duality between spaces and algebras of functions on these spaces, the idea in noncommutative geometry is to forget the commutativity of the algebras of functions and replace them by appropriate classes of noncommutative associative algebras. In this talk we present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang--Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical ``quaternionic torus” SU(2) x SU(2) in parallel with the action of the torus U(1) x U(1) on a ``complex” noncommutative torus.