Hopf brace, braid equation and bicrossed coproduct

Abstract

In this paper, we mainly give some equivalent characterisations of Hopf braces, show that the category CB(A) of Hopf braces is equivalent to the category C(A) of bijective 1-cocycles, and prove that the category CB(A) of Hopf braces is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many more Hopf braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel'd doubles of finite dimensional Hopf algebras, and give a sufficient and necessary condition for a given bicrossed coproduct A ⊲⊳ H to be a Hopf brace if A or H is a Hopf brace.

Rota—Baxter groups, skew left braces, and the Yang—Baxter equation

“…A definition of RB-operators on arbitrary Hopf algebras can be found in [3]. In [30], Hopf braces were defined and studied. It was shown in [7] that some operators on the Sweedler algebra H 4 , satisfying only the condition (1.1), are RB-operators on the adjoint Lie algebra H algebra.…”