[Paper Review] Generalized Lie bialgebras and Jacobi structures on Lie groups

David Iglesias, Juan Carlos Marrero|ArXiv.org|Feb 21, 2001

Advanced Topics in Algebra16 references4 citations

TL;DR

This paper introduces generalized Lie bialgebras as the infinitesimal invariants of Lie groups equipped with special Jacobi structures, extending the classical Lie bialgebra framework. It establishes a correspondence between generalized Lie bialgebras and Jacobi structures on Lie groups, generalizes the Yang-Baxter equation method for construction, and fully classifies compact generalized Lie bialgebras, showing that non-trivial 1-cocycles only occur for su(2).

ABSTRACT

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propose a method to obtain generalized Lie bialgebras. It is a generalization of the Yang-Baxter equation method. Finally, we describe the structure of a compact generalized Lie bialgebra.

Motivation & Objective

• To establish a correspondence between generalized Lie bialgebras and Jacobi structures on Lie groups, generalizing the known link between Lie bialgebras and Poisson Lie groups.
• To extend the Yang-Baxter equation method for constructing Lie bialgebras to the generalized setting, enabling new constructions of generalized Lie bialgebras.
• To classify compact generalized Lie bialgebras with non-trivial 1-cocycles, particularly identifying when such structures exist and their underlying Lie algebraic constraints.
• To explore the relationship between algebraic Jacobi structures on Lie algebras and generalized Lie bialgebras, especially in the context of contact and locally conformal symplectic structures.

Proposed method

• Define generalized Lie bialgebras as pairs $((\mathfrak{g}, \phi_0), (\mathfrak{g}^*, X_0))$ where $\mathfrak{g}$ is a Lie algebra, $\phi_0 \in \mathfrak{g}^*$ is a 1-cocycle, and $X_0 \in \mathfrak{g}$ is a 1-cocycle in the dual, satisfying compatibility conditions.
• Use the Schouten-Nijenhuis bracket to define Jacobi structures on Lie groups via the pair $(\Lambda, E)$ satisfying $[\Lambda, \Lambda] = 2E \wedge \Lambda$ and $[E, \Lambda] = 0$.
• Construct generalized Lie bialgebras via a generalized Yang-Baxter equation method, extending the classical method for Lie bialgebras.
• Apply the theory to algebraic Jacobi structures on Lie algebras, particularly focusing on contact and locally conformal symplectic structures.
• Use representation-theoretic and cohomological techniques to analyze compact generalized Lie bialgebras, especially leveraging $\mathrm{ad}$-invariant inner products.
• Prove that any compact Lie algebra with a non-trivial algebraic contact structure must be isomorphic to $\mathfrak{su}(2)$, using the maximality of the characteristic direction.

Experimental results

Research questions

• RQ1 How can generalized Lie bialgebras be characterized as the infinitesimal invariants of Lie groups endowed with Jacobi structures?
• RQ2 What is the generalized Yang-Baxter equation method for constructing generalized Lie bialgebras, and how does it extend the classical construction?
• RQ3 Which compact Lie algebras admit non-trivial generalized Lie bialgebra structures, and what are the constraints on the 1-cocycles $\phi_0$ and $X_0$?
• RQ4 What is the precise relationship between algebraic Jacobi structures on a Lie algebra and the resulting generalized Lie bialgebra?
• RQ5 Can all compact generalized Lie bialgebras with non-trivial 1-cocycles be classified, and what is the role of the Lie algebra structure in this classification?

Key findings

• Generalized Lie bialgebras are the infinitesimal invariants of Lie groups equipped with Jacobi structures, generalizing the Lie bialgebra–Poisson Lie group correspondence.
• The generalized Yang-Baxter equation method allows for the systematic construction of generalized Lie bialgebras from algebraic Jacobi structures on Lie algebras.
• Any compact Lie algebra admitting a non-trivial generalized Lie bialgebra structure with $\phi_0 \neq 0$ or $X_0 \neq 0$ must be isomorphic to $\mathfrak{su}(2)$.
• All algebraic contact structures on $\mathfrak{su}(2)$ arise from a 1-form $\eta = \mu_1 e^1 + \mu_2 e^2 + \mu_3 e^3$ with $\mu = (\mu_1, \mu_2, \mu_3) \neq 0$, and the associated generalized Lie bialgebra is determined by $r = \lambda^1 e_2 \wedge e_3 - \lambda^2 e_1 \wedge e_3 + \lambda^3 e_1 \wedge e_2$, $X_0 = - (\lambda^1 e_1 + \lambda^2 e_2 + \lambda^3 e_3)$ with $\lambda^i = -\mu_i / (\mu_1^2 + \mu_2^2 + \mu_3^2)$.
• For a compact Lie algebra $\mathfrak{h}$ of dimension $2k+1$, if it admits a non-trivial algebraic contact structure, then $k=1$ and $\mathfrak{h} \cong \mathfrak{su}(2)$, with the characteristic direction generating a maximal abelian subalgebra.
• The existence of a non-trivial 1-cocycle in the generalized Lie bialgebra setting implies that the Lie algebra must be semisimple and of rank 1, restricting the possible compact examples to $\mathfrak{su}(2)$.