$\ddot{I}‰$-Lie bialgebras and $\ddot{I}‰$-Yang-Baxter equation

Abstract

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

23 pages