Let \(R\colon V\otimes V\to V\otimes V\) be a linear operator, where \(V\) is a vector space over a field \(k\). The authors use the notation \(R^{12}=R\otimes I\), \(R^{23}=I\otimes R\), \(R^{13}=(I\otimes T)(R\otimes I)(I\otimes T)\), where \(T\colon V\otimes V\to V\otimes V\) denotes the twist operator and \(I\) denotes the identity map. An invertible linear operator \(R\) is called a Yang-Baxter operator if it satisfies the equation \(R^{12}\circ R^{23}\circ R^{12}=R^{23}\circ R^{12}\circ R^{23}\).
Let \(A\) be an associative \(k\)-algebra. The first section is devoted to characterize those parameters \(\alpha,\beta,\gamma\in k\) such that the linear map \(\varphi^A_{\alpha,\beta,\gamma}\colon A\otimes A\to A\otimes A\) defined by \(\varphi^A_{\alpha,\beta,\gamma}(a\otimes b)=\alpha ab\otimes 1+\beta 1\otimes ab-\gamma a\otimes b\) is a YB operator (Theorem 1.1). In the second section it is shown that a YB operator of the type \(\varphi^A_{\alpha,\beta,\gamma}\), where \(A=C^*\) is the dual algebra for a coalgebra \(C\), induces a YB operator \(\psi\colon C\otimes C\to C\otimes C\). In the third section, the YB operators obtained in the previous sections are classified. Two examples are given in the last section.