\(R\)-matrices for Leibniz algebras (Q1395670)

The authors introduce \(R\)-matrices for Leibniz algebras as a direct generalization of the classical \(R\)-matrices. The linear mapping \(R_{\pm}: L\to L\) of a Leibniz algebra \(L\) is called an \(R_{\pm}\)-matrix if the new bilinear operator defined by \([X,Y]_{R_{\pm}}=[RX,Y]\pm [X,RY]\), \(X,Y\in L\), satisfies the Jacobi-Leibniz identity. The authors prove that \(R\) is an \(R_{\pm}\)-matrix if it satisfies a Yang-Baxter type equation. In the concluding remarks they outline possible relations of their constructions to integrable systems.