Decomposable plane curves of degree up to 5 were shown to be quantum homogeneous spaces by Brown and Tabiri. It was conjectured that all decomposable plane curves of any degree are quantum homogeneous spaces. In this talk, we will discuss recent results which show that decomposable surfaces and plane curves of any degree are quantum homogeneous spaces. Other algebras such as the reduced algebra will be constructed and its properties discussed.

I will discuss certain relations between 3-dimensional topological gauge theories with continuous and finite gauge groups, commonly known as Chern–Simons and Dijkgraaf–Witten theories respectively. The relations of this form appear when the continuous and finite gauge groups are the same algebraic group considered over the complex/real numbers and a finite field, respectively. In this talk, I will focus on the SU(2) example and consider the relationship on the level of the corresponding invariants of closed 3-manifolds: Witten–Reshetikhin–Turaev and Dijkgraaf–Witten invariants.