Various doublings of Hopf Algebras. Algebras of Operators on Quantum Groups and Complex Cobordisms

Abstract

Family of doublings of Hopf algeras based on the product of algebra and its dual are constructed and studied. Special cases of these construction may be considered as natural quantum analogs of rings of differential operators on groups. Such constructions appeared in 1966-67 in the authors works in the complex cobordism theory. These constructions do not leed to the Hopf algebras (except of the special case of the Drinfeld's quantum double which is not the same as the natural quantum analog of the ring of operators on groups). However, they have important ''almost Hopf'' properties important in particular in the topological and analytical applications.