Extensions of Hopf Algebras and Lie Bialgebras

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