On the braided structures of bicrossproduct Hopf algebras

Abstract

In this paper we show that if $ H\star$ $A$ is a bicrossproduct Hopf algebra then $(H\star A, \sigma)$ is braided if and only if $\sigma$ has a unique form: $\sigma(h\otimes a, g\otimes b)=\sum\beta(h_{1}, g_{1})\omega(h_{2}, g_{2(-1)})\omega(h_{3}, b_{1})\alpha(a_{1}, b_{2})\tau(a_{2}, g20)$ such that $\beta$, $\omega$, $\tau$ and $\alpha$ satisfy certain compatible conditions. The result is applied to a certain bicrossproduct of $H$ and $H^{cop}$, where $H$ is a Hopf algebra with bijective antipode.