Classifying complements for Hopf algebras and Lie algebras

Let A E be a given extension of Hopf (respectively Lie) algebras. We answer the 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 HA2(H,A). 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 4n2-dimensional Hopf algebra whose factorization index over the group algebra is arbitrary large.