Factorization of quasitriangular structures of smash biproduct bialgebras
Fujun Wang
TL;DR
This work establishes a general factorization theory for quasitriangular structures on smash biproduct bialgebras $A{_\tau\times_\sigma}B$ under the right conormal condition on $\sigma$. It introduces normalized factor elements $W,X,Y,Z$ extracted from any quasitriangular structure and proves that a QT-structure exists if and only if these factors satisfy a finite set of identities $\text{C1–C19}$, enabling a explicit reconstruction of the $R$-matrix. The framework unifies and extends prior results for Radford's biproducts, bicross products, and dual double cross products, and shows how extra normal/conormal hypotheses yield simpler, known factorization formulas (MW, ZZ, J2). The results provide a cohesive method to classify and construct quasitriangular smash biproducts, with implications for quantum Yang–Baxter equations and related invariants in Hopf-algebra theory.
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.