Factorisation structures of algebras and coalgebras

TL;DR

This paper investigates the factorisation problem for bialgebras, establishing conditions under which a bialgebra can be expressed as a product of algebra and coalgebra components, and introduces a new product construction that generalizes previous work.

Contribution

The paper introduces a new product construction for algebras and coalgebras that characterizes when a bialgebra factorizes into such components, generalizing known constructions.

Findings

Characterizes when a bialgebra factorizes as a product of algebra and coalgebra.

Introduces the product $L_W\bowtie_R H$ with necessary and sufficient conditions for bialgebra structure.

Recovers and generalizes constructions by Majid, Radford, and recent pointed Hopf algebras.

Abstract

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.