Quantum Symmetries in Noncommutative Geometry

Abstract

Abstract: In this thesis, we study quantum symmetries within the realm of noncommutative geometry. These symmetries are captured in two levels of generality, namely, Hopf algebras (or compact quantum groups) in the context of noncommutative differential geometry a la Connes and Hopf algebroids in the context of noncommutative Kaehler geometry a la Ó Buachalla. We compute the orientation-preserving quantum isometry group of the Chakraborty-Pal spectral triple on the odd sphere. Generalizing the Hopf algebra case, we build a framework to take into account Hopf algebroid equivariance in non-commutative complex geometry. We classify complex structures on a canonical spectral triple over the three-point space and identify a universal Hopf algebroid acting on a finite space.