$\omega$-Lie bialgebras and $\omega$-Yang-Baxter equation

In this paper, we introduce the definition of multiplicative $\omega$-Lie bialgebra, which is equivalent to the Manin triples and matched pairs. We also study the $\omega$-Yang-Baxter equation and Yang-Baxter $\omega$-Lie bialgebra. The skew-symmetric solutions of the $\omega$-Yang-Baxter equation can be used to construct Yang-Baxter $\omega$-Lie bialgebra. We further introduce the concept of the $\omega$-$\mathcal{O}$-operator, which can be constructed from a left-symmetric algebras, and based on the $\omega$-$\mathcal{O}$-operator, we construct skew-symmetric solutions to the $\omega$-Yang--Baxter equation.