Leibniz Bialgebras, Classical Yang–Baxter Equations and Dynamical Systems

Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory

“…Recently, the notion of anti-pre-Lie bialgebras was studied in [23], which serves as a preliminary to supply a reasonable bialgebra theory for transposed Poisson algebras [24]. The notions of mock-Lie bialgebras [25] and Leibniz bialgebras [26,27] were also introduced with different motivations. These bialgebras have a common property in that they can be equivalently characterized by Manin triples which correspond to nondegenerate invariant bilinear forms on the algebra structures.…”

Pre-Leibniz algebras

“…By substituting ( 23), ( 24) and ( 25) in the identity (22), we get that (a, {µ k } k≥1 ) is a Leibniz ∞ -algebra.…”

“…In [20] Loday and Pirashvili introduced a cohomology theory for Leibniz algebras with coefficients in a representation. Given a vector space g, Balavoine [4] constructs a graded Lie algebra (known as Balavoine's graded Lie algebra) on the space of multilinear maps on g whose Maurer-Cartan elements correspond to Leibniz algebra structures on g. Recently, Rota-Baxter operators and relative Rota-Baxter operators on Leibniz algebras are introduced and their relation with Leibniz Yang-Baxter equation and Leibniz bialgebras are discovered in [22,24,25]. Like a Rota-Baxter operator on a Lie algebra induces a pre-Lie algebra structure, a (relative) Rota-Baxter operator on a Leibniz algebra gives rise to a pre-Leibniz algebra (already introduced in [24] by the name of Leibniz-dendriform algebra) structure.…”