[Paper Review] S-expansions of three-dimensional Lie algebras

Maryna Nesterenko|arXiv (Cornell University)|Dec 8, 2012

Advanced Topics in Algebra1 references3 citations

TL;DR

This paper investigates S-expansions of three-dimensional real Lie algebras, demonstrating that unimodular Lie algebras (like sl(2,ℝ)) can be S-expanded into non-unimodular ones (e.g., A₃.₃), and that such expansions can connect algebras in both directions. Despite this, S-expansions do not impose an ordering on the variety of Lie algebras of a fixed dimension, limiting their utility for classifying solvable Lie algebras via expansion from semisimple ones.

ABSTRACT

S-expansions of three-dimensional real Lie algebras are considered. It is shown that the expansion operation allows one to obtain a non-unimodular Lie algebra from a unimodular one. Nevertheless S-expansions define no ordering on the variety of Lie algebras of a fixed dimension.

Motivation & Objective

• To examine whether S-expansions can generate all solvable three-dimensional Lie algebras from semisimple ones, revisiting Zaitsev's conjecture.
• To determine if S-expansions can produce non-unimodular Lie algebras from unimodular ones, challenging the assumption that unimodularity is preserved under expansion.
• To investigate whether S-expansions establish a hierarchical or ordering relationship among Lie algebras of fixed dimension.
• To analyze the structural properties preserved or altered under S-expansion and subalgebra extraction, particularly solvability and nilpotency degrees.
• To explore the limitations of S-expansions in classifying solvable Lie algebras by comparing forward and reverse expansion paths.

Proposed method

• S-expansion uses an Abelian semigroup S and a Lie algebra 𝔤 to construct a new Lie algebra 𝔖×𝔤 with structure constants defined by the semigroup multiplication: C_{(i,α)(j,β)}^{(k,γ)} = C_{ij}^k if λ_αλ_β = λ_γ, and 0 otherwise.
• Resonance decomposition is applied to extract smaller-dimensional Lie algebras from the expanded algebra by partitioning the semigroup and Lie algebra into subspaces with compatible multiplication and bracket closure.
• Reduced algebras are constructed via projection: if 𝔖×𝔤 = 𝔠𝔤 ⊕ 𝔥𝔤 with [𝔠𝔤, 𝔥𝔤] ⊂ 𝔥𝔤, then 𝔠𝔤 inherits a Lie bracket from the projection of the original bracket.
• Explicit S₃-expansions of sl(2,ℝ) and A₂.₁⊕A₁ are computed to generate nine-dimensional algebras, from which three-dimensional subalgebras isomorphic to A₃.₃ are extracted.
• Basis changes are used to verify isomorphisms, such as transforming [E₁,E₆]=E₁, [E₂,E₆]=E₂ into the standard form of A₂.₁⊕A₁.
• The paper compares solvability and nilpotency degrees before and after expansion, showing they are preserved under S-expansion and subalgebra extraction.

Experimental results

Research questions

• RQ1 Can S-expansions generate non-unimodular Lie algebras from unimodular ones, thereby challenging the unimodularity preservation of contractions?
• RQ2 Does the S-expansion procedure establish a partial order on the variety of Lie algebras of a fixed dimension, as suggested by Zaitsev’s conjecture?
• RQ3 Can all three-dimensional solvable Lie algebras be obtained via S-expansion from semisimple or simple Lie algebras of the same dimension?
• RQ4 What structural properties (e.g., solvability degree, center, Cartan subalgebra) are preserved or altered under S-expansion and subalgebra extraction?
• RQ5 Why are certain non-unimodular algebras like A₃.₂, A₃.₄ᵃ, and A₃.₅ᵇ inaccessible via S-expansion from simple Lie algebras?

Key findings

• S-expansion of sl(2,ℝ) via S₃ produces a nine-dimensional algebra from which a three-dimensional subalgebra isomorphic to A₃.₃ is extracted, demonstrating that unimodular algebras can generate non-unimodular ones.
• The inverse process—S₂-expansion of A₃.₃—recovers A₂.₁⊕A₁, showing that S-expansion can connect algebras in both directions, thus breaking any potential ordering.
• The solvability and nilpotency degrees of the original and expanded algebras are preserved under S-expansion and subalgebra extraction.
• The center dimension decreases from 1 to 0, the Cartan subalgebra dimension from 2 to 1, and the derived algebra dimension from 1 to 2 after S-expansion from A₂.₁⊕A₁ to A₃.₃.
• It is conjectured that A₃.₂, A₃.₄ᵃ, and A₃.₅ᵇ cannot be obtained via S-expansion from simple Lie algebras due to incompatibility in their commutation relations, though a rigorous proof is still needed.
• S-expansions do not define a partial order on the variety of Lie algebras of fixed dimension, as demonstrated by bidirectional connections between A₂.₁⊕A₁ and A₃.₃.