Extensions of Hopf algebras and Lie bialgebras

Let f, g be finite-dimensional Lie algebras over a field of characteristic zero. Regard f and g*, the dual Lie coalgebra of g, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair (f, g*) of Lie bialgebras is given, which has structure maps -->, rho. Then it induces a matched pair (Uf, Ug(o), -->', rho') of Hopf algebras, where Uf is the universal envelope of f and U g(o) is the Hopf dual of Ug. We show that the group Opext(Uf, Ug(o)) of cleft Hopf algebra extensions associated with (Uf, U g(o), -->', rho') is naturally isomorphic to the group Opext(f, g*) of Lie bialgebra extensions associated with (f, g*, -->, rho). An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If g = [g, g], there follows a bijection between the set Ext(Uf, Ug(o)) of all cleft Hopf algebra extensions of Uf by Ug(o) and the set Ext(f, g*) of all Lie bialgebra extensions of f by g*.