Matched pairs and Yang-Baxter operators
Yunnan Li
TL;DR
The paper studies how matched pairs of actions on a Hopf algebra $H$ and Yetter-Drinfeld braces yield braiding and Yang-Baxter operators. It establishes equivalent descriptions: a matched pair $(H,\rightharpoonup,\leftharpoonup)$ corresponds to a braiding operator $r$ on $H$, and the operator is involutive precisely when the intrinsic Hopf algebra $H_{\rightharpoonup}$ in ${_H^H}{\mathcal{Y}\mathcal{D}}$ is braided-commutative, together with antipode compatibilities $S(x\leftharup y)=S(y)\rightharpoonup S(x)$ and $S(x\rightharup y)=S(y)\leftharup S(x)$. The work proves a fundamental isomorphism $H\bowtie H \cong H_{\rightharpoonup}\# H$ (bosonization) and gives a constructive criterion for involutivity via these structures. An explicit classification on the $8$-dimensional Hopf algebra $A_{C_2\times C_2}$ yields two families of matched pairs whose associated Yang-Baxter operators are involutive, including a coquasitriangular-structure–derived family and a non-coquasitriangular one. Altogether, the results extend the LYZ/AGV/GGV framework to non-cocommutative settings and provide concrete classifications and involutivity criteria for YB operators arising from matched pairs of actions.
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 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 $H_\rightharpoonup$ in the category of Yetter-Drinfeld modules over $H$ is braided commutative. Also, we show that the double cross product $H\bowtie H$ is a Hopf algebra with a projection and $H_\rightharpoonup$ serves as its subalgebra of coinvariants. As an illustration, we use a simplified characterization to classify matched pairs of actions on the 8-dimensional non-semisimple Hopf algebra $A_{C_2\times C_2}$ and analyze the associated Yang-Baxter operators to find that they are all involutive.