Extensions of Hopf Algebras and Lie Bialgebras

Abstract:

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 $\mathfrak {g} ^*$.