Ado's theorem

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.

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).

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.

• {{Citation | last=Ado | first=Igor D. | title=Note on the representation of finite continuous groups by means of linear substitutions|journal= Izv. Fiz.-Mat. Obsch. (Kazan')|volume= 7 |year=1935|pages=1–43}}. (Russian language)
• {{Citation | last=Ado | first=Igor D. | title=The representation of Lie algebras by matrices | url=http://mi.mathnet.ru/eng/umn/v2/i6/p159 | language=Russian | mr=0027753 | year=1947 | journal=Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=2 | issue=6 | pages=159–173}} translation in {{Citation | last1=Ado | first1=Igor D. | title=The representation of Lie algebras by matrices | mr=0030946 | year=1949 | journal=American Mathematical Society Translations | issn=0065-9290 | volume=1949 | issue=2 | pages=21}}
• {{Citation | last1=Iwasawa | first1=Kenkichi | authorlink=Kenkichi Iwasawa| title=On the representation of Lie algebras | mr=0032613 | year=1948 | journal=Japanese Journal of Mathematics | volume=19 | pages=405–426}}
• {{Citation | last=Harish-Chandra | authorlink=Harish-Chandra | title=Faithful representations of Lie algebras | jstor=1969352 | mr=0028829 | year=1949 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=50 | pages=68–76 | doi=10.2307/1969352}}
• {{Citation | last=Hochschild | first=Gerhard |authorlink=Gerhard Hochschild| title=An addition to Ado's theorem | year=1966 | journal=Proceedings of the American Mathematical Society | volume=17 | pages=531–533 | url=https://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0194482-0/home.html | doi=10.1090/s0002-9939-1966-0194482-0| doi-access=free }}
• Nathan Jacobson, Lie Algebras, pp. 202–203

• [http://terrytao.wordpress.com/2011/05/10/ados-theorem/ Ado’s theorem], comments and a proof of Ado's theorem in Terence Tao's blog What's new.

Category:Lie algebras

Category:Theorems about algebras