English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We establish a bialgebra structure on Rota-Baxter Lie algebras following theManin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebrasare characterized by generalizing matched pairs of Lie algebras and Manintriples of Lie algebras to the context of Rota-Baxter Lie algebras. Thecoboundary case leads to the introduction of the admissible classicalYang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which theantisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of$\mathcal{O}$-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Liealgebras are introduced to produce antisymmetric solutions of the admissibleCYBE. Furthermore, extending the well-known property that a Rota-Baxter Liealgebra of weight zero induces a pre-Lie algebra, the Rota-Baxter Lie bialgebraof weight zero induces a bialgebra structure of independent interest, namelythe special L-dendriform bialgebra, which is equivalent to a Lie group with aleft-invariant flat pseudo-metric in geometry. This induction is alsocharacterized as the inductions between the corresponding Manin triples andmatched pairs. Finally, antisymmetric solutions of the admissible CYBE in aRota-Baxter Lie algebra of weight zero give special L-dendriform bialgebras. Inparticular, both Rota-Baxter algebras of weight zero and Rota-Baxter pre-Liealgebras of weight zero can be used to construct special L-dendriform algebras.

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