  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.
