On cross product Hopf algebras

TL;DR

This paper investigates conditions under which cross product structures on algebras and coalgebras in braided monoidal categories form bialgebras or Hopf algebras, providing explicit descriptions of related projections.

Contribution

It establishes necessary and sufficient conditions for cross product algebras and coalgebras to form bialgebras or Hopf algebras, extending the understanding of their structure.

Findings

Criteria for $A imes B$ to be a bialgebra or Hopf algebra

Explicit descriptions of Hopf algebra projections in various cases

Conditions under which cross product Hopf algebras are double cross, biproduct, or smash products

Abstract

Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for $A\ot B$ to be a bialgebra, and sufficient conditions for $A\ot B$ to be a Hopf algebra. We discuss when such a cross product Hopf algebra is a double cross (co)product, a biproduct, or, more generally, a smash (co)product Hopf algebra. In each of these cases, we provide an explicit description of the associated Hopf algebra projection.