[Paper Review] Nambu-Lie Groups

Izu Vaisman|arXiv (Cornell University)|Dec 10, 1998

Advanced Topics in Algebra11 references18 citations

TL;DR

This paper introduces Nambu-Lie groups as Lie groups equipped with a multiplicative Nambu structure, defined by the condition that the Nambu tensor vanishes at the identity and the Nambu bracket of left- or right-invariant 1-forms remains invariant. It establishes a corresponding Nambu-Lie algebra and provides multiple examples, extending Poisson-Lie theory to higher-order Nambu structures.

ABSTRACT

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several examples of Nambu-Lie groups and algebras.

Motivation & Objective

• To extend the concept of Poisson-Lie groups to higher-order Nambu structures by defining Nambu-Lie groups.
• To characterize Nambu-Lie groups via the vanishing of the Nambu tensor at the identity and invariance of the Nambu bracket on left/right-invariant 1-forms.
• To define a corresponding algebraic structure, the Nambu-Lie algebra, as the infinitesimal counterpart of Nambu-Lie groups.
• To provide explicit examples of Nambu-Lie groups and algebras to illustrate the theory.

Proposed method

• Extend the Nambu bracket from functions to differential 1-forms to define a compatible tensor structure on Lie groups.
• Define a Nambu-Lie group as a Lie group with a multiplicative Nambu tensor P satisfying P(e) = 0, where e is the group identity.
• Impose the condition that the Nambu bracket of left-invariant 1-forms is itself left-invariant, and similarly for right-invariant forms.
• Derive the infinitesimal version of the structure, leading to the definition of a Nambu-Lie algebra as the tangent algebra at the identity.
• Use the invariance properties to classify and construct examples of Nambu-Lie groups and algebras.
• Verify that the multiplicative and invariance conditions are equivalent to the Nambu tensor being compatible with the group multiplication.

Experimental results

Research questions

• RQ1 What conditions must a Lie group satisfy to admit a multiplicative Nambu structure?
• RQ2 How can the Poisson-Lie framework be generalized to Nambu structures of order greater than two?
• RQ3 What is the infinitesimal counterpart of a Nambu-Lie group, and how is it characterized algebraically?
• RQ4 Which Lie groups admit nontrivial Nambu-Lie structures, and what are their structural properties?
• RQ5 How do left- and right-invariant 1-forms behave under the Nambu bracket in a Nambu-Lie group?

Key findings

• A Lie group is a Nambu-Lie group if and only if the Nambu tensor vanishes at the identity and the Nambu bracket of left-invariant 1-forms is left-invariant.
• The Nambu bracket of right-invariant 1-forms is right-invariant under the same conditions, ensuring full multiplicative compatibility.
• The infinitesimal counterpart of a Nambu-Lie group is a Nambu-Lie algebra, defined by the same invariance and vanishing conditions at the Lie algebra level.
• Several examples of Nambu-Lie groups and algebras are explicitly constructed, demonstrating the existence and diversity of such structures.
• The framework generalizes Poisson-Lie theory to higher-order Nambu structures while preserving key invariance and multiplicativity properties.