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

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

Índice

## Dichiarazione[]

Ado's theorem states that every finite-dimensional Lie algebra l over a field K di characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. Più precisamente, il teorema lo afferma l ha un linear representation ρ over K, on a finite-dimensional vector space V, that is a faithful representation, fabbricazione l isomorphic to a subalgebra of the endomorphisms di V.

## Storia[]

The theorem was proved in 1935 di Igor Dmitrievich Ado di 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).

## Implicazioni[]

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.

## Riferimenti[]

• 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 russo), 2 (6): 159–173, ISSN 0042-1316, SIG 0027753 translation in Ado, Igor D. (1949), "The representation of Lie algebras by matrices", American Mathematical Society Translations, 1949 (2): 21, ISSN 0065-9290, SIG 0030946
• Iwasawa, Kenkichi (1948), "On the representation of Lie algebras", Japanese Journal of Mathematics, 19: 405–426, SIG 0032613
• Harish-Chandra (1949), "Faithful representations of Lie algebras", Annali di matematica, Seconda serie, 50: 68–76, doi:10.2307/1969352, ISSN 0003-486X, JSTOR 1969352, SIG 0028829
• Hochschild, Gerhard (1966), "An addition to Ado's theorem", Atti dell'American Mathematical Society, 17: 531–533, doi:10.1090/s0002-9939-1966-0194482-0
• Nathan Jacobson, Algebre di bugia, pp. 202–203