Ado's theorem

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 true in general), 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.

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

External links

* [http://eom.springer.de/l/l058590.htm Springer Encyclopedia page]
* [http://mathworld.wolfram.com/AdosTheorem.html Page at MathWorld]