Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras

Abstract

We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples of Lie algebras to the context of Rota-Baxter Lie algebras. The coboundary case leads to the introduction of the admissible classical Yang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which the antisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of O-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Lie algebras are introduced to produce antisymmetric solutions of the admissible CYBE. Furthermore, extending the well-known property that a Rota-Baxter Lie algebra of weight zero induces a pre-Lie algebra, the Rota-Baxter Lie bialgebra of weight zero induces a bialgebra structure of independent interest, namely the special L-dendriform bialgebra, which is equivalent to a Lie group with a left-invariant flat pseudo-metric in geometry. This induction is also characterized as the inductions between the corresponding Manin triples and matched pairs. Finally, antisymmetric solutions of the admissible CYBE in a Rota-Baxter Lie algebra of weight zero give special L-dendriform bialgebras and in particular, both Rota-Baxter algebras of weight zero and Rota-Baxter pre-Lie algebras of weight zero can be used to construct special L-dendriform bialgebras.

Bialgebras, the Yang–Baxter equation and Manin triples for mock-Lie algebras

“…Such a Lie algebra together with its additional structure is called a Lie bialgebra. So a bialgebra structure on a given algebra is obtained by a corresponding set of comultiplication together with the set of compatibility conditions between multiplication and comultiplication [12]. For example, take a finite dimensional vector space V with a given algebraic structure, this can be acheived by equipping the dual space V * with the same algebraic structure and a set of compatibility conditions between the structures on V and those on V * .…”

“…Many scientists have found solutions for the Yang-Baxter equation, however the full classification of its solutions remains an open problem. In the theory of Lie bialgebras, it is essential to consider the coboundary case, which is related to the theory of the classical Yang-Baxter equation [12,10,8,30]. We aim to have an analogue in the mock-Lie case.…”

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

“…Moreover, bialgebra theory and the classical Yang-Baxter equation for Hom-Lie algebra version were established in [28]. Recently, Rota-Baxter Lie bialgebras and endo Lie bialgebras were studied in [2,4,17]. Chen, Stiénon, and Xu examined weak Lie 2-bialgebras by using big brackets with respect to which S • (V [2] ⊕ V * [1]) is a graded Lie algebra, and proved that (strict) Lie 2-bialgebras are in one-one correspondence with crossed modules of Lie bialgebras ( [11]).…”

“…Recently, Rota-Baxter Lie bialgebras and endo Lie bialgebras were studied in [2,4,17]. Chen, Stiénon, and Xu examined weak Lie 2-bialgebras by using big brackets with respect to which S • (V [2] ⊕ V * [1]) is a graded Lie algebra, and proved that (strict) Lie 2-bialgebras are in one-one correspondence with crossed modules of Lie bialgebras ( [11]). Moreover, they proved that there is a one-to-one correspondence between connected, simply-connected (quasi-)Poisson Lie 2-groups and (quasi-)Lie 2-bialgebras in [12].…”