Quantum differential calculus on graphs

Quantum groups or Hopf algebras are an exciting new generalisation of ordinary groups. They have a rich mathematical structure and numerous roles in situations where ordinary groups are not adequate. The goal in this thesis is to set out this mathematical structure by proceeding gradually from the basic concepts to the more advanced ones. Furthermore, we extend the idea of classical differential geometry to a noncommutative world. We start the thesis by presenting some basic algebraic structures, in particular algebras and coalgebras, to arrive to the notion of Hopf algebras. We continue by presenting the definition of universal enveloping algebra U(g) of a Lie group g. After that, we give a Hopf algebra structure on U(g) and we analyze the Hopf duality relationship between the Hopf algebras U(sl2) and O(SL2). Then, we go quantum and we investigate the Hopf algebras Uq(sl2) and Oq(SL2) and their duality relationship. In the second part of the thesis, we develop a theory of differential geometry over noncommutative algebras. It will be possible to study the first order differential calculus Ω1 on the algebra of functions defined on a directed graph with a finite set of vertices. Finally, some algorithms used in the field of Geometric Deep Learning on graphs for node classification are presented. The concepts discussed are then applied in the creation of an architecture capable of classifying the nodes of the Zachary Karate Club dataset.