Compatible Left-Symmetric Bialgebras

A compatible left-symmetric algebra is a pair of left-symmetric algebras satisfying that any linear combination of the two left-symmetric products is a left-symmetric product. We construct a bialgebra theory of compatible left-symmetric algebras as an analogue of a Lie bialgebra. They can also be regarded as a “compatible version” of left-symmetric bialgebras, that is, a pair of left-symmetric bialgebras satisfying that any linear combination of the two left-symmetric bialgebras is still a left-symmetric bialgebra. Many properties of compatible left-symmetric bialgebras as the “compatible version” of the corresponding properties of left-symmetric bialgebras are presented. In particular, there is a coboundary compatible left-symmetric bialgebra theory and an [Formula: see text]-equation, which is an analogue of the classical Yang-Baxter equations in Lie algebras.