[Paper Review] On Nambu-Lie 3-algebra representations

Corneliu Sochichiu|ArXiv.org|Jun 23, 2008

Advanced Topics in Algebra26 references14 citations

TL;DR

This paper proposes a method to construct matrix representations of Nambu–Lie 3-algebras by leveraging irreducible representations of their underlying Lie algebras, obtained by fixing one generator. For the four-dimensional Euclidean 3-algebra 𝒜₄, the construction reveals that unitary representations require matrices with complex eigenvalues—specifically fourth roots of −1—making purely Hermitian matrix representations impossible, despite the algebra's Euclidean signature.

ABSTRACT

We propose a recipe to construct matrix representations of Nambu--Lie 3-algebras in terms of irreducible representations of underlying Lie algebra. The case of Euclidean four-dimensional 3-algebra is considered in details. We find that representations of this 3-algebra are not possible in terms of only Hermitian matrices in spite of its Euclidean nature.

Motivation & Objective

• To develop a systematic method for constructing matrix representations of Nambu–Lie 3-algebras using irreducible representations of their underlying Lie algebras.
• To investigate whether the four-dimensional Euclidean 3-algebra 𝒜₄ admits unitary representations in terms of Hermitian matrices.
• To explore the role of the fixed generator (e.g., T₄) in inducing a Lie algebra structure and its impact on representation constraints.
• To determine whether non-constant or non-commuting representations of the fixed generator can circumvent the impossibility of Hermitian realizations.
• To clarify the general structural features of 3-algebra representations, particularly the necessity of complex eigenvalues in compact cases.

Proposed method

• Define a Nambu–Lie 3-algebra via a totally anti-symmetric 3-bracket with structure constants satisfying the Fundamental Identity.
• Fix one generator (e.g., T₄) to induce a Lie algebra structure on the remaining generators via the bracket [T₄, Tᵢ, Tⱼ] = i fᵢⱼₖ Tₖ.
• Construct matrix representations by requiring the Nambu commutator [A,B,C]_N = ABC + BCA + CAB − BAC − ACB − CBA to match the 3-bracket relations.
• Assume the fixed generator T₄ is diagonalizable in the representation, reducing the Nambu bracket to a deformed Lie bracket depending on its eigenvalues.
• Use irreducible representations of the underlying su(2) algebra to build solutions, expanding matrices in Pauli matrices to solve the deformed commutation relations.
• Analyze both constant and non-constant T₄ cases, showing that non-commuting T₄ leads to a consistent deformation of the Lie bracket, but ultimately reduces to the same complex eigenvalue condition.

Experimental results

Research questions

• RQ1 Can the four-dimensional Euclidean Nambu–Lie 3-algebra 𝒜₄ be represented using only Hermitian matrices?
• RQ2 What constraints does the underlying Lie algebra structure impose on the representation of a Nambu–Lie 3-algebra?
• RQ3 Is it possible to construct a unitary representation of 𝒜₄ using non-constant or non-commuting generators for the fixed element T₄?
• RQ4 Why do representations of compact Nambu–Lie 3-algebras necessarily require complex eigenvalues, even in Euclidean signature?
• RQ5 How does the uplift procedure from Lie algebras relate to the metric signature and the possibility of Hermitian representations?

Key findings

• The four-dimensional Euclidean 3-algebra 𝒜₄ cannot be represented using only Hermitian matrices, despite its positive-definite metric.
• All unitary representations of 𝒜₄ require generators with eigenvalues that are fourth roots of −1, implying complex matrices are necessary.
• The construction of representations via irreducible representations of the underlying Lie algebra (e.g., su(2)) leads to a consistent but complex eigenvalue structure.
• Non-constant T₄ representations, though more general, still reduce to the same eigenvalue condition, confirming that complex eigenvalues are unavoidable.
• The requirement for complex eigenvalues in compact 3-algebras contrasts sharply with standard Lie algebra representations, where Hermitian generators are common.
• The mechanism of sign flip in the metric (as in Lorenzian 3-algebras) may be analogous to the complex eigenvalue condition in Euclidean 3-algebras, suggesting a deeper structural link.