[Paper Review] Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras

Chengming Bai, Li Guo+1 more|arXiv (Cornell University)|Apr 20, 2016

Advanced Topics in Algebra21 references9 citations

TL;DR

This paper introduces two new types of 3-Lie bialgebras—local cocycle and double construction 3-Lie bialgebras—extending Lie bialgebra theory to 3-Lie algebras. It establishes connections to the 3-Lie classical Yang-Baxter equation, $\/mathcal{O}$-operators, and 3-pre-Lie algebras, showing that solutions of the 3-Lie CYBE yield coboundary local cocycle 3-Lie bialgebras, while double construction bialgebras give rise to pseudo-metric 3-Lie algebras with neutral signature via Manin triples.

ABSTRACT

This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are therefore called the local cocycle 3-Lie bialgebra and the double construction 3-Lie bialgebra. They can be regarded as suitable extensions of the well-known Lie bialgebra in the context of 3-Lie algebras, in two different directions. The local cocycle 3-Lie bialgebra is introduced to extend the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a ternary variation of the classical Yang-Baxter equation, called the 3-Lie classical Yang-Baxter equation, a ternary $\mathcal{O}$-operator and a 3-pre-Lie algebra is established. In particular, it is shown that solutions of the 3-Lie classical Yang-Baxter equation give (coboundary) local cocycle 3-Lie bialgebras, whereas 3-pre-Lie algebras give rise to solutions of the 3-Lie classical Yang-Baxter equation. The double construction 3-Lie bialgebra is introduced to extend to the 3-Lie algebra context the connection between Lie bialgebras and double constructions of Lie algebras. Their related Manin triples give a natural construction of pseudo-metric 3-Lie algebras with neutral signature. Moreover, the double construction 3-Lie bialgebra can be regarded as a special class of the local cocycle 3-Lie bialgebra. Explicit examples of double construction 3-Lie bialgebras are provided.

Motivation & Objective

• To extend Lie bialgebra theory to 3-Lie algebras by introducing two distinct bialgebra structures: local cocycle and double construction 3-Lie bialgebras.
• To establish a connection between solutions of the 3-Lie classical Yang-Baxter equation and coboundary local cocycle 3-Lie bialgebras.
• To generalize the double construction of Lie bialgebras to the 3-Lie algebra setting, constructing pseudo-metric 3-Lie algebras with neutral signature via Manin triples.
• To clarify the relationship between the two bialgebra types, showing that double construction 3-Lie bialgebras are a special case of local cocycle 3-Lie bialgebras.

Proposed method

• Introduce the local cocycle 3-Lie bialgebra using a compatibility condition defined by a 1-cocycle in the tensor representation of the adjoint representation.
• Define the 3-Lie classical Yang-Baxter equation (3-Lie CYBE) and show that its solutions yield coboundary local cocycle 3-Lie bialgebras.
• Introduce $\mathcal{O}$-operators in the 3-Lie context and prove that 3-pre-Lie algebras give rise to solutions of the 3-Lie CYBE.
• Construct double construction 3-Lie bialgebras via matched pairs of 3-Lie algebras and Manin triples, ensuring compatibility between the algebra and coalgebra structures.
• Use the dual of the comultiplication to define a 3-Lie algebra structure on the dual space, forming a Manin triple.
• Provide explicit examples, including a non-trivial double construction 3-Lie bialgebra on the 4-dimensional simple 3-Lie algebra, using a symmetric $r$-matrix not satisfying the 3-Lie CYBE.

Experimental results

Research questions

• RQ1 How can Lie bialgebra theory be extended to 3-Lie algebras through compatible bialgebra structures?
• RQ2 What is the role of the 3-Lie classical Yang-Baxter equation in constructing coboundary local cocycle 3-Lie bialgebras?
• RQ3 How do $\mathcal{O}$-operators and 3-pre-Lie algebras relate to solutions of the 3-Lie CYBE?
• RQ4 Can double construction 3-Lie bialgebras be constructed such that their Manin triples yield pseudo-metric 3-Lie algebras with neutral signature?
• RQ5 Under what conditions does a non-trivial double construction 3-Lie bialgebra exist, and why do some 3-Lie algebras only admit the trivial (zero coproduct) case?

Key findings

• Solutions of the 3-Lie classical Yang-Baxter equation give rise to coboundary local cocycle 3-Lie bialgebras.
• 3-pre-Lie algebras provide a construction method for solutions of the 3-Lie classical Yang-Baxter equation.
• Double construction 3-Lie bialgebras are a special class of local cocycle 3-Lie bialgebras, with the compatibility condition expressed via a 1-cocycle and skew-symmetry.
• The Manin triple associated with a double construction 3-Lie bialgebra yields a pseudo-metric 3-Lie algebra with neutral signature.
• For the 4-dimensional simple complex 3-Lie algebra, a non-trivial double construction 3-Lie bialgebra exists, with a symmetric $r$-matrix not satisfying the 3-Lie CYBE.
• In several 3-Lie algebra classes (e.g., 3D non-trivial, 4D classes (2), (5), (6)), only the trivial double construction bialgebra (zero coproduct) exists due to inconsistency between 1-cocycle and skew-symmetry.