On a class of Lie groups with a left invariant flat pseudo-metric

Bai, Chengming; Hou, Dongping; Chen, Zhiqi

doi:10.1007/s00605-010-0207-y

Cited by 5 publications

(19 citation statements)

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“…Preferably, such a Rota-Baxter operator on Lie bialgebras should be part of a package of Rota-Baxter type actions applied systematically throughout the diagram in (5). This is what we will achieve, as depicted in the following diagram with the corresponding Lie bialgebra boldfaced in both diagrams.…”

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confidence: 90%

“…This paper establishes a bialgebra structure on Rota-Baxter Lie algebras following the approach of Manin triples in the classical work on Lie bialgebras [17] and the recent development on Rota-Baxter antisymmetric infinitesimal bialgebras [6]. The well-known connection of Rota-Baxter Lie algebras with pre-Lie algebras is lifted to the bialgebra level, establishing a relationship of Rota-Baxter Lie bialgebras with special L-dendriform bialgebras [5]. 1.1.…”

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confidence: 97%

“…Since a Rota-Baxter operator on a Lie algebra induces a pre-Lie algebra, a Rota-Baxter operator on a matched pair or Manin triple of Lie algebras should induce a matched pair or Manin triple of pre-Lie algebras. Quite unexpectedly to us, the resulting bialgebra structure is not a pre-Lie bialgebra as introduced in [3], but a special L-dendriform bialgebra [5], leading to the commutative double square diagram in (6).…”

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confidence: 98%

“…Most of the results in [6] for Rota-Baxter associative algebras remain valid for Rota-Baxter Lie algebras. Therefore by adding the roles of Rota-Baxter operators in diagram (5), we get the left part of the diagram (6).…”

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confidence: 99%

“…To determine the correct induced structure from a Rota-Baxter Lie bialgebra, we regard the latter as a Lie bialgebra with a Rota-Baxter operator. Since a Lie bialgebra is characterized by a matched pair or a Manin triple of Lie algebras, as shown in the right column in the diagram (5), we can also regard a Rota-Baxter Lie bialgebra as a matched pair or Manin triple of Lie algebras equipped with a Rota-Baxter operator. Since a Rota-Baxter operator on a Lie algebra induces a pre-Lie algebra, a Rota-Baxter operator on a matched pair or Manin triple of Lie algebras should induce a matched pair or Manin triple of pre-Lie algebras.…”

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

Bai¹,

Guo²,

Liu³

et al. 2022

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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.

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confidence: 90%

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confidence: 97%

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confidence: 98%

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confidence: 99%

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confidence: 99%

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