Quasi-triangular, triangular, factorizable anti-Leibniz bialgebras and anti-Leibniz Yang-Baxter equation

We introduce the notion of an anti-Leibniz bialgebra which is equivalent to a Manin triple of anti-Leibniz algebras, is equivalent to a matched pair of anti-Leibniz algebras. The study of some special anti-Leibniz bialgebras leads to the introduction of the anti-Leibniz Yang-Baxter equation in an anti-Leibniz algebra. A symmetric (or an invariant) solution of the anti-Leibniz Yang-Baxter equation gives an anti-Leibniz bialgebra. The notion of a relative Rota-Baxter operator of an anti-Leibniz algebra is introduced to construct symmetric solutions of the anti-Leibniz Yang-Baxter equation. Moreover, we introduce the notions of factorizable anti-Leibniz bialgebras and skew-symmetric Rota-Baxter anti-Leibniz algebras, and show that a factorizable anti-Leibniz bialgebra leads to a factorization of the underlying anti-Leibniz algebra. There is a one-to-one correspondence between factorizable anti-Leibniz bialgebras and skew-quadratic Rota-Baxter anti-Leibniz algebras. Finally, we constrict anti-Leibniz bialgebras form Leibniz bialgebras by the tensor product and constrict infinite-dimensional anti-Leibniz bialgebras form finite-dimensional anti-Leibniz bialgebras by the completed tensor product.