Ado's Theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional representation, meaning it can be realized concretely as an algebra of matrices under the commutator bracket. Named for Igor Ado, it guarantees that the abstract axioms defining a Lie algebra never describe anything more general than the matrix examples that originally motivated the theory, so every finite-dimensional Lie algebra can in principle be studied as a Lie algebra of matrices.
Facts
StatementEvery finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation as a Lie algebra of square matrices under the commutator bracket, that is, it is isomorphic to a subalgebra of the endomorphisms of some finite-dimensional vector space. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Ado's Theorem (Wikipedia)
Sources
1. Ado's Theorem (Wikipedia)
Wikimedia FoundationHistory section, first sentence
The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov.
In Branch: Algebra, Lead sentence
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
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