An analogue of the classical Yang-Baxter equation for general algebraic structures
The author introduces an analogue of the classical Yang-Baxter equation for a general complex nonassociative algebra \({\mathcal A}\) with operation `\(\cdot\)'. Under certain conditions this equation is equivalent to the equation \[ R(z_1+ z_2) (x\cdot R( -z_1) (y))+ R(z_2) (R(z_1) (x)\cdot y)= R(z_1+ z_2) (x)\cdot R(z_2) (y) \] for \(x\), \(y\) in \({\mathcal A}\) and \(z_1, z_2\) in \(\mathbb{C}\), where the solution \(R\) is an analytic function with values in \(\text{End } {\mathcal A}\). The author gives several ways to construct solutions in the case when \({\mathcal A}\) is graded by an abelian group. As an example the author gives some unitary nondegenerate trigonometric solutions of the classical Yang-Baxter equation for affine Lie algebras. It is noted that the above equation can be modified for the case when \({\mathcal A}\) is a vertex operator algebra and that solutions can be constructed in a similar way.
scientific article; zbMATH DE number 757762
scientific article