Classifying complements for Hopf algebras and Lie algebras

Let A⊆EA⊆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|(▹,◃))HA2(H,A|(▹,◃)), where (▹,◃)(▹,◃) is the matched pair associated to H  . The factorization index [E:A]f[E:A]f is introduced as a numerical measure of the (CCP). For two n  -th roots of unity we construct a 4n24n2-dimensional Hopf algebra whose factorization index over the group algebra is arbitrary large.