abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, Ado's theorem is a theorem characterizing finite-dimensional

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

Statement

Ado's theorem states that every finite-dimensional

''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 In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

''V'', that is a

faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group (mathematics), group on a vector space is a linear representation in which different elements of are represented by ...

, making ''L'' isomorphic to a subalgebra of the

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

s of ''V''.

History

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).

Implications

While for the Lie algebras associated to

classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...

s there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

''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 In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a ...

References

* . (Russian language) * translation in * * * {{Citation , last=Hochschild , first=Gerhard , authorlink=Gerhard Hochschild, title=An addition to Ado's theorem , year=1966 , journal=

Proceedings of the American Mathematical Society ''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles ...

, 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

External links

Ado’s theorem
comments and a proof of Ado's theorem in Terence Tao's blog ''What's new''. Lie algebras Theorems about algebras