Ado's theorem

In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.

Statement

Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that L has a linear representation ρ over K, on a finite-dimensional vector space V, that is a faithful representation, making L isomorphic to a subalgebra of the endomorphisms of V.

History

The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov.
The restriction on the characteristic was later removed by Kenkichi Iwasawa.

Implications

While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group G, it does not imply that G has a faithful linear representation, but rather that G always has a linear representation that is a local isomorphism with a linear group.