[Paper Review] A construction of finite-dimensional faithful representation of Lie algebra

Yury A. Neretin|arXiv (Cornell University)|Feb 19, 2002

Advanced Topics in Algebra11 citations

TL;DR

This paper presents a natural, constructive proof of the Ado theorem by building a finite-dimensional faithful representation for any finite-dimensional Lie algebra through a semidirect product structure of a reductive algebra and a nilpotent ideal. The construction uses the enveloping algebra of the nilpotent part, quotients by a suitable ideal to ensure finiteness, and leverages derivations induced by the reductive part to define the action, yielding a faithful module.

ABSTRACT

We give a natural proof of the Ado theorem.

Motivation & Objective

• To provide a natural, constructive proof of the Ado theorem, which asserts the existence of a finite-dimensional faithful representation for any finite-dimensional Lie algebra.
• To eliminate the reliance on the Chevalley construction of the algebraic envelope by directly proving a key lemma enabling the semidirect product decomposition.
• To establish a systematic method for embedding any Lie algebra into a semidirect product of a reductive algebra and a nilpotent ideal with completely reducible action.
• To introduce the concept of elementary expansions as a tool to iteratively build such embeddings through codimension-1 ideals.
• To demonstrate that the faithful representation can be explicitly constructed using derivations on the enveloping algebra modulo a carefully chosen ideal.

Proposed method

• Construct a finite-dimensional faithful representation by considering a Lie algebra 𝔤 as a semidirect product 𝔭 ⋉ 𝔫, where 𝔭 is reductive and 𝔫 is nilpotent with faithful adjoint action.
• Use the universal enveloping algebra U(𝔫) as the representation space, equipped with left multiplication by 𝔫 and derivation actions by 𝔭 via the Leibniz rule.
• Define an ideal I in U(𝔫) generated by all products of length greater than k+2, where k is the nilpotency class of 𝔫, to ensure finiteness.
• Form the quotient space U(𝔫)/I, which inherits a finite-dimensional 𝔤-module structure via the actions of 𝔫 and 𝔭.
• Prove that I ∩ A = 0, where A is the linear span of 1, x, and x₁x₂, ensuring the quotient is finite-dimensional and the action is faithful.
• Use elementary expansions—introducing formal generators y and z with specific commutation rules based on the Jordan–Chevalley decomposition of a derivation—to inductively build the required semidirect product structure.

Experimental results

Research questions

• RQ1 Can the Ado theorem be proven constructively without relying on the algebraic envelope construction?
• RQ2 How can a finite-dimensional faithful representation of a Lie algebra be explicitly constructed using the structure of its Levi decomposition?
• RQ3 What conditions ensure that the action of a reductive subalgebra on a nilpotent ideal is completely reducible and faithful?
• RQ4 Can the embedding of a Lie algebra into a semidirect product of reductive and nilpotent subalgebras be achieved through a finite sequence of elementary expansions?
• RQ5 What role does the Jordan–Chevalley decomposition of a derivation play in constructing faithful representations via enveloping algebras?

Key findings

• The construction yields a finite-dimensional faithful representation of any finite-dimensional Lie algebra by forming the quotient U(𝔫)/I, where I is the ideal generated by products of length > k+2.
• The action of 𝔤 = 𝔭 ⋉ 𝔫 on U(𝔫)/I is well-defined and faithful, as the intersection I ∩ A = 0 ensures non-degeneracy.
• The ideal I is invariant under the derivations d_z induced by 𝔭, preserving the module structure.
• The method relies on the existence of a reductive subalgebra 𝔭 and a nilpotent ideal 𝔫 such that the adjoint action of 𝔭 on 𝔫 is faithful and completely reducible.
• Lemma 1, which guarantees the existence of such a decomposition, is proven directly without invoking the Chevalley construction.
• Elementary expansions allow iterative construction of the required semidirect product structure, reducing the problem step-by-step until the desired embedding is achieved.