Factorization of quasitriangular structures of smash biproduct bialgebras

TL;DR

This paper characterizes when smash biproduct bialgebras are quasitriangular by identifying specific elements and identities, generalizing previous results for various types of Hopf algebras.

Contribution

It provides a new factorization criterion for quasitriangular structures in smash biproduct bialgebras, extending earlier work on related algebraic structures.

Findings

Characterization of quasitriangular smash biproduct bialgebras

Existence of normalized elements satisfying specific identities

Generalization of previous results on Hopf algebra structures

Abstract

In this paper, we consider the factorization and reconstruction of quasitriangular structures of smash biproduct bialgebras. Let $A{_\tau\times_\sigma}B$ be a smash biproduct bialgebra. Under condition that $\sigma$ is right conormal, we prove that $A{_\tau\times_\sigma}B$ 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 $A{_\tau\times_\sigma}B$ is given as $\sum Z {^1_{\tau_1\tau_2}}\bar{X}{^1_{\tau_3}}X^1\otimes W^1Y^1\otimes Z^2 Y{^2_{\sigma_1\sigma_2}}\epsilon_B(1_{B\tau_1\sigma_2} \bar{X}{^2_{\sigma_1}})\otimes1_{B\tau_2}1_{B\tau_3}X^2W^2$. 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.