Ado's Theorem

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In abstract algebra, **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*.

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. It was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov.

The restriction on the characteristic was removed later, by Iwasawa and Harish-Chandra.

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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

The restriction on the characteristic was removed later, by Iwasawa and Harish-Chandra.

- I. D. Ado,
*Note on the representation of finite continuous groups by means of linear substitutions*, Izv. Fiz.-Mat. Obsch. (Kazan') , 7 (1935) pp. 1–43 (Russian language) - translation in
- Nathan Jacobson,
*Lie Algebras*, pp. 202-203

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