In this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge...Show moreIn this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge theory, in which we identify the moduli algebra with the function algebra on the moduli space of flat connections. Secondly, the group gauge theory case will be extended to the group algebra case, for which we show that the quantum moduli algebra is isomorphic to the moduli algebra in the group gauge theory case. Thirdly, we will give a general construction of how to obtain quantum moduli algebras of semisimple finite-dimensional Hopf algebras, and we will identify this construction with the construction of the quantum moduli algebra in the group algebra case. Fourthly, we will be examining the situation in which our Hopf algebra is the Drinfel'd double of a group algebra. After giving some examples, we will show that the quantum moduli algebra in the case of the Drinfel'd double is isomorphic to the quantum moduli algebra in the group algebra case.Show less
