abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, 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 iden ...

Statement

Ado's theorem states that every finite-dimensional

''L'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' of

characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...

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 Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

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 on a vector space is a linear representation in which different elements of are represented by distinct linear ...

, 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 ...

s of ''V''.

History

The theorem was proved in 1935 by

Igor Dmitrievich Ado Igor Dmitrievich Ado (Russian: Игорь Дмитриевич Адо, scientific transliteration ''Igor' Dmitrievič Ado''; January 17, 1910 in KazanJune 30, 1983) was a Soviet mathematician. He was born into the family of a state employee and ...

Kazan State University Kazan (Volga region) Federal University (russian: Казанский (Приволжский) федеральный университет, tt-Cyrl, Казан (Идел буе) федераль университеты) is a public research uni ...

, a student of

Nikolai Chebotaryov Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev, uk, Мико́ла Григо́рович Чеботарьо́в, russian: Никола́й Григо́рьевич Чеботарёв) ( – 2 July 1947) was a Ukrainia ...

. The restriction on the characteristic was later removed by

Kenkichi Iwasawa Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. Biography Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gun ...

(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 , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...

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 that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

''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 Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administra ...

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 f ...

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. As a requirement, all articles must be at most 15 printed pages. According to the ...

, 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 Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...

, ''Lie Algebras'', pp. 202–203

External links

Ado’s theorem
comments and a proof of Ado's theorem in

Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

's blog ''What's new''. Lie algebras Theorems about algebras