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 Sources
1. Ado's Theorem (Wikipedia)
Wikimedia FoundationHistory section, first sentenceQuote, History section, first sentence
The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.