# 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 (see also the below Gerhard Hochschild paper for a proof).

## 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 (which is not true in general), but rather that *G* always has a linear representation that is a local isomorphism with a linear group.

## References

- Ado, Igor D. (1935), "Note on the representation of finite continuous groups by means of linear substitutions",
*Izv. Fiz.-Mat. Obsch. (Kazan')*,**7**: 1–43. (Russian language) - Ado, Igor D. (1947), "The representation of Lie algebras by matrices",
*Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk*(in Russian),**2**(6): 159–173, ISSN 0042-1316, MR 0027753 translation in Ado, Igor D. (1949), "The representation of Lie algebras by matrices",*American Mathematical Society Translations*,**1949**(2): 21, ISSN 0065-9290, MR 0030946 - Iwasawa, Kenkichi (1948), "On the representation of Lie algebras",
*Japanese Journal of Mathematics*,**19**: 405–426, MR 0032613 - Harish-Chandra (1949), "Faithful representations of Lie algebras",
*Annals of Mathematics*, Second Series,**50**: 68–76, doi:10.2307/1969352, ISSN 0003-486X, JSTOR 1969352, MR 0028829 - Hochschild, Gerhard (1966), "An addition to Ado's theorem",
*Proceedings of the American Mathematical Society*,**17**: 531–533, doi:10.1090/s0002-9939-1966-0194482-0 - Nathan Jacobson,
*Lie Algebras*, pp. 202–203

## External links

- Ado’s theorem, comments and a proof of Ado's theorem in Terence Tao's blog
*What's new*.