Solution of the Yang-Baxter equation not coming from quasi-triangular structure

Question

Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}.$$ Does $R$ come from a quasi-triangular structure on $A$? That is, does it exist a bialgebra structure on $A$ such that the pair $(A,R)$ is a quasi-triangular bialgebra?