Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras

By Chengming Bai, Li Guo, Xianglong Ni. 2009

159 citations recorded by the source corpus.

We generalize the classical study of (generalized) Lax pairs, the related $${\mathcal O}$$ -operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended $${\mathcal O}$$ -operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized r-matrix ansatz and the related double Lie algebra structures. Relationship between extended $${\mathcal O}$$ -operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain an explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind $${\mathcal O}$$ -operators and fits in a setup of the triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a5776d2f2a2c25fad863d1a64de56055de49bb9c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Affine actions on Lie groups and post-Lie algebra structures (2011, result): “een discussed in several articles of Loday, see for example [17] and the references given therein. Furthermore, post-Lie algebras also turned up in relation with the classical Yang-Baxter equation in [1]. In this paper, we will show that these algebras also naturally appear in yet another context, n”
Rota-Baxter operators on cocommutative Hopf algebras. (2020, unknown): “for all x, y ∈ g [17].”
Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras (2019, methodology): “AS, DENDRIFORM ALGEBRAS AND PRE-LIE ALGEBRAS 21 This completes the proof. 4.2. MatchingPostLiealgebras. ThenotionofaPostLiealgebraarisingfromanoperadstudy[42] and ﬁnding quite broad applications. See [7] for example. We now introduce the concept of a matching Lie algebra based on which we propose an”