Self-duality and generalized Bicrossproducts Hopf algebras

Group factorizations are very common in mathematics. Among their uses is the bicrossproduct construction which is one of the primary sources of non-commutative and noncocommutative Hopf algebras. These bicrossproduct Hopf algebras have been introduced by Majid10 and Takeuchi16. Since then, bicrossproduct Hopf algebras have been extensively studied2-4,6,9. These algebras have many applications, for example Majid in10 showed that they can be considered as a systems combine quantum mechanics with geometry11.