Factorization of quasitriangular structures of smash biproduct bialgebras

Abstract

In this paper, we consider the factorization and reconstruction of quasitriangular structures of smash biproduct bialgebras. Let $A{_τ\times_σ}B$ be a smash biproduct bialgebra. Under condition that $σ$ is right conormal, we prove that $A{_τ\times_σ}B$ is quasitriangular if and only if there exists a set of normalized elements $W\in B\otimes B$, $X\in A\otimes B$, $Y\in B\otimes A$ and $Z\in A\otimes A$ satisfying a certain series of identities. In this case, the quasitriangular structure of $A{_τ\times_σ}B$ is given as $\sum Z {^1_{τ_1τ_2}}\bar{X}{^1_{τ_3}}X^1\otimes W^1Y^1\otimes Z^2 Y{^2_{σ_1σ_2}}ε_B(1_{Bτ_1σ_2} \bar{X}{^2_{σ_1}})\otimes1_{Bτ_2}1_{Bτ_3}X^2W^2$. Our result generalizes the similar results for Radford's biproduct Hopf algebras studied by L. Zhao and W. Zhao, for bicrossproduct Hopf algebras studied by Zhao, Wang and Jiao, and for the dual Hopf algebras of double cross product Hopf algebras studied by Jiao.