Program 2018

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

We will discuss a double geometry formulation of the isotropic rigid rotator, as a dynamical system whose configuration space is usually chosen to be the group manifold of SU(2). I will generalize the carrier space of the dynamics in terms of the double group of SU(2) and discuss the emerging geometry. The construction is thus extended to principal chiral models as an instance of $1+1$ field theory with non-Abelian duality.

Con metodo delle orbite quantico si intende una corrispondenza tra foglie simplettiche di una varietà di Poisson e *-rappresentazioni irriducibili della sua algebra quantizzante. Mostreremo come l'approccio alla quantizzazione di spazi omogenei di Poisson tramite la integrazione simplettica permetta di costruire esplicitamente tale corrispondenza e quali informazioni aggiuntive (di natura topologica) possa dare

In noncommutative geometry Hopf-Galois extensions of algebras are regarded as principal bundles, with Hopf algebras playing the role of structure groups. This seminar will present results, obtained in collaboration with P. Aschieri and G. Landi, on the study of the group of gauge transformations in this algebraic framework

The notion of noncommutative principal bundle is not as well understood as
that of noncommutative vector bundle. Main examples, like the
noncommutative instanton bundle, are algebraically understood as Hopf-Galois
extensions. This definition is particularly useful when the noncommutative
base space is affine (the function algebra being given in terms of
generators and relations). We relax it by presenting a sheaf theoretic
approach that allows to consider noncommutative principal bundles over non
affine base spaces; examples include projective spaces.