Matched pairs and Yang–Baxter operators

Abstract Recently, Ferri and Sciandra introduced two equivalent algebraic structures, matched pair of actions on an arbitrary Hopf algebra and Yetter–Drinfeld brace. In fact, they equivalently produce braiding operators on Hopf algebras satisfying the braid equation, thus generalize the construction of Yang–Baxter operators by Lu, Yan and Zhu from braiding operators on groups, and also by Angiono, Galindo and Vendramin from cocommutative Hopf braces. In this paper, we first provide equivalence conditions for such kind of Yang–Baxter operators to be involutive. Particularly, we give a positive answer for an open problem raised by Ferri and Sciandra, namely, a matched pair of actions on a Hopf algebra H induces an involutive Yang–Baxter operator if and only if its intrinsic Hopf algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mo>⇀</m:mo> </m:msub> </m:math> {H_{\rightharpoonup}} in the category of Yetter–Drinfeld modules over H is braided commutative. Next we show that the double cross product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>⋈</m:mo> <m:mi>H</m:mi> </m:mrow> </m:math> {H\bowtie H} is a Hopf algebra with a projection, whose subalgebra of coinvariants is isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mo>⇀</m:mo> </m:msub> </m:math> {H_{\rightharpoonup}} . It indicates that double cross product with a projection is also equivalent to matched pair of actions on a Hopf algebra.