Classifying complements for Hopf algebras and Lie algebras

Let $A\subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given complement then all the other complements are obtained from $H$ by a certain type of deformation. We establish a bijective correspondence between the isomorphism classes of all complements of $A$ in $E$ and a cohomological type object ${\mathcal H}{\mathcal A}^{2} (H, A , | , (\triangleright, \triangleleft) )$,where$(\triangleright, \triangleleft)$ is the matched pair associated to $H$. The factorization index $[E:A{]}^f$ is introduced as a numerical measure of the (CCP). For two $n$-th roots of unity we construct a $4n^2$-dimensional Hopf algebra whose factorization index over the group algebra is arbitrary large.