Factorisation structures of algebras and coalgebras

We consider the factorisation problem for bialgebras: when a bialgebra $K$ factorises as $K=HL$, where $H$ and $L$ are algebras and coalgebras (but not necessarly bialgebras). Given two maps $R: H\ot L\to L\ot H$ and $W:L\ot H\to H\ot L$, we introduce a product $L_W\bowtie_R H$ and give necessary and sufficient conditions for $L_W\bowtie_R H$ to be a bialgebra. It turns out that $K$ factorises as $K=HL$ if and only if $K\cong L_W\bowtie_R H$ for some maps $R$ and $W$. As examples of this product we recover constructions introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed bu Beattie, D\u asc\u alescu and Gr\"unenfelder appear as special cases.