Extensions of Hopf Algebras and Lie Bialgebras

Abstract

Let f \mathfrak {f} , g \mathfrak {g} be finite-dimensional Lie algebras over a field of characteristic zero. Regard f \mathfrak {f} and g ∗ \mathfrak {g} ^* , the dual Lie coalgebra of g \mathfrak {g} , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair ( f , g ∗ ) (\mathfrak {f} , \mathfrak {g} ^*) of Lie bialgebras is given, which has structure maps ⇀ , ρ \rightharpoonup , \rho . Then it induces a matched pair ( U f , U g ∘ , ⇀ ′ , ρ ′ ) (U\mathfrak {f}, U\mathfrak {g}^{\circ },\rightharpoonup ’, \rho ’) of Hopf algebras, where U f U\mathfrak {f} is the universal envelope of f \mathfrak {f} and U g ∘ U\mathfrak {g}^{\circ } is the Hopf dual of U g U\mathfrak {g} . We show that the group O p e x t ( U f , U g ∘ ) \mathrm {Opext} (U\mathfrak {f},U\mathfrak {g}^{\circ }) of cleft Hopf algebra extensions associated with ( U f , U g ∘ , ⇀ ′ , ρ ′ ) (U\mathfrak {f}, U\mathfrak {g} ^{\circ }, \rightharpoonup ’, \rho ’ ) is naturally isomorphic to the group Opext ⁡ ( f , g ∗ ) \operatorname {Opext}(\mathfrak {f},\mathfrak {g} ^*) of Lie bialgebra extensions associated with ( f , g ∗ , ⇀ , ρ ) (\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 g = [ g , g ] \mathfrak {g} =[\mathfrak {g} , \mathfrak {g}] , there follows a bijection between the set E x t ( U f , U g ∘ ) \mathrm {Ext}(U\mathfrak {f} , U\mathfrak {g}^{\circ }) of all cleft Hopf algebra extensions of U f U\mathfrak {f} by U g ∘ U\mathfrak {g}^{\circ } and the set E x t ( f , g ∗ ) \mathrm {Ext}(\mathfrak {f}, \mathfrak {g}^*) of all Lie bialgebra extensions of f \mathfrak {f} by g ∗ \mathfrak {g} ^* .