Ado's theorem
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In mathematics, 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 shows not that G has a faithful linear representation (which is not a valid theorem), but 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.
[edit] References
- 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)
- I. D. Ado, The representation of Lie algebras by matrices" Transl. Amer. Math. Soc. (1) , 9 (1962) pp. 308–327 Uspekhi Mat. Nauk. , 2 (1947) pp. 159–173
- K. Iwasawa, On the representation of Lie algebras, Japanese Journal of Mathematics, vol. 19 (1948), pp. 405-426
- Harish-Chandra, Faithful representations of Lie algebras. Ann. Math. 50 (1949) 68-76
- Nathan Jacobson, Lie Algebras, pp. 202-203
