Leibniz-dendriform bialgebras and relative Rota-Baxter operators

Abstract: In this paper, we introduce the notion of Leibniz-dendriform bialgebras and show that it is equivalent to phase spaces and matched pairs of Leibniz algebras. The study of coboundary Leibniz-dendriform bialgebras naturally leads to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We show that skew-symmetric solutions of the LD-YBE give rise to such coboundary bialgebras. Furthermore, we investigate how solutions without the skew-symmetry condition can also induce Leibniz-dendriform bialgebras. This motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we prove that solutions of the LD-YBE whose symmetric parts are invariant yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras.