HQFTs and Quantum Groups

The thesis is divided into two parts one for dimension 2 and the other for dimension 3. Part one of the thesis generalizes the denition of an n-dimensional HQFT in terms of a monoidal functor from a rigid symmetric monoidal category XCobn 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 ane case and conjectured for non-ane 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.

Autorentext

Neha Gupta is doing M. Tech. (Computer Science and Engineering) from The NorthCap University, Gurugram. She received her MCA degree from YMCA Institute of Engineering, Faridabad in 2007.