Lyubashenko V. Squared Hopf algebras and the modular functor

The thesis is devoted to squared, braided and modular Hopf algebras and modular categories. The reconstruction theorems for (monoidal, rigid, braided and ribbon) Abelian categories are proved. The criterion of modularity of the category of modules over a finite-dimensional ribbon Hopf algebra (quantum group) is found. The existence and uniqueness of an integral in braided Hopf algebras is proved. A squared and a braided Hopf algebras are canonically associated with a modular category. Their modular properties are proved, and, as a corollary, representations of modular groups of surfaces and invariants of the closed three-dimensional varieties are constructed.