Unifying Radford's biproducts over Hopf braces (Q6637177)

According to [\textit{I. Angiono} et al., Proc. Am. Math. Soc. 145, No. 5, 1981--1995 (2017; Zbl 1392.16032)], a \textit{Hopf brace} over a coalgebra \(\left(H,\Delta, \varepsilon\right)\) consists of the following data:\N\N(1)\ A Hopf algebra structure \(\left(H, \cdot, 1, \Delta, \varepsilon, S\right)\);\N\N(2)\ a Hopf algebra structure \(\left(H,\circ , 1_{\circ}, \Delta, \varepsilon, T \right)\) satisfying the identity \N\[\N\forall\, g, h, l\in H\quad g\circ\left(hl\right) = \left(g_1\circ h\right)S\left(g_2\right)\left(g_3\circ l\right). \N\]\NIn [\textit{H. Zhu} and \textit{Z. Ying}, Commun. Algebra 50, No. 4, 1426--1440 (2022; Zbl 1505.16047)], a Hopf brace version of ``Radford's biproduct'' theorem is introduced. In particular, it is based on Radford's biproduct of Hopf algebras, which is significant in the classification of pointed Hopf algebras, see [\textit{N. Andruskiewitsch} and \textit{H.-J. Schneider}, Ann. Math. (2) 171, No. 1, 375--417 (2010; Zbl 1208.16028)].\N\NIn the paper under review, the authors show that, if on the one hand, Radford's biproduct theorem on Hopf braces allows for constructing several instances of pointed Hopf braces, on the other some new examples of pointed Hopf braces cannot be obtained by the same theorem. Motivated by this fact, they find some generalized Radford's biproduct theorem on Hopf braces to classify pointed Hopf braces. Specifically, they use some braided tensor category from Hopf braces (constructed in [\textit{H. Zhu}, Linear Multilinear Algebra 70, No. 16, 3171--3188 (2022; Zbl 1510.16030)]) to present some unifying Radford's biproducts over Hopf braces.