Hqfts and Quantum Groups: Homotopy Quantum Field Theory (Hqft) in Dimensions Two and Three and Quantum Groups over Group-schemes

The thesis is divided into two parts one for dimension 2 and the other for dimension 3. Part one of the thesis generalizes the de?nition of an n-dimensional HQFT in terms of a monoidal functor from a rigid symmetric monoidal category X?Cobn to any monoidal category A. In particular, 2-dimensional HQFTs with target K(G,1) taking values in A are generated from any Turaev G-crossed system in A and vice-versa. This is the generalization of Turaev's theory into a purely categorical set-up. Part two of the thesis generalizes the concept of a group-coalgebra, Hopf group-coalgebra, crossed Hopf group-coalgebra and quasitriangular Hopf group-coalgebra over a group scheme. Quantum double of a crossed Hopf group-scheme coalgebra is constructed in the a?ne case and conjectured for non-a?ne case. We can construct 3-dimensional HQFTs from modular crossed G-categories. The category of representations of a quantum double of a crossed Hopf group-coalgebra is a ribbon(quasitriangular) crossed group-category. Hence generates 3-dimensional HQFTs under certain conditions if the category becomes modular. However, systematic ?nding of modular crossed G-categories is largely open.