The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

摘要

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R: H direct X L -> L direct X H and W: L direct X H -> H direct X L. We introduce a product K = L _W(infinity)_R H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dascalescu and Grunenfelder appear as special cases.