Bialgebras structure and splitting of operations for averaging mock-Lie algebras

Averaging mock-Lie algebras are mock-Lie algebras equipped with averaging operators. In this paper we establish a bialgebra structure on averaging mock-Lie algebras following the Manin triple approach to mock-Lie bialgebras. Explicitely averaging mock-Lie bialgebras are characterized by generalizing matched pairs of mock-Lie algebras and Manin triple of mock-Lie algebras to the concept of averaging mock-Lie algebras. The coboundary case leads to the introduction of admissible mock-Lie-Yang-Baxter equation (mLYBe) in averaging mock-Lie algebras, for which the antisymetrique solutions give rise to averaging mock-Lie bialgebras. Furthermore, the notion of O-operator on averaging mock-Lie algebras and averaging pre-mock-Lie algebras are introduced in relation with structures of anti-Leibniz algebras.