Lie group - Lie group

Japanese: リー群 - りーぐん（英語表記）Lie group

A connected analytic manifold G is called a Lie group or analytic _group _if it is also a group, the group operation ( g1 , g2 ₎ → g1 _g2 is an analytic mapping from the product manifold G × G to G , and the inverse operation g → g - ¹ is an analytic mapping from G to G.

Thus, a Lie group is a concept that combines manifolds and groups, and was devised by the Norwegian mathematician M. S. Lee in order to study geometry and differential equations using group theory, and so it is called a Lie group after him.

The Lie group G is also called a real Lie group or a complex Lie group depending on whether it is a real or complex manifold. The additive group R of all real numbers and the real general linear group GL ( n , R ) are real Lie groups, while the additive group C of all complex numbers and the complex general linear group GL ( n , C ) are complex Lie groups.

As in the case of abstract groups, subgroups, normal subgroups, quotient groups, homomorphisms, isomorphisms, etc. of Lie groups have been defined taking into account their natural conditions as manifolds, and the theory of Lie groups, which is similar to group theory, has been completed.

A distinctive feature of Lie group theory is that the Lie algebra ɡ is naturally constructed for a Lie group G and plays an important role, as will be described below.

For an element g of a Lie group G , there is an analytic map Φ _g : G ∋ x → gx ∈ G
The differential dΦ _g of gives a linear space isomorphism from the tangent space T ( e ) at the identity e of G onto the tangent space T ( g ) at g . Now, for Xe _∈ T ( e ),
X : G ∋ g → dΦ _g・( Xe )∈ T ( g )
Then, we create a vector field X on the manifold G. The set of such X 's ɡ={ X | X _e ∈ T ( e )} is a linear space isomorphic to T ( e ), and the product of vector fields 〔 X , Y 〕= X・Y ― Y・X
This algebra ɡ is written as L ( G ) and is called the Lie algebra of the Lie group G.

The Lie algebra of the Lie group GL ( n , R ) is the Lie algebra ɡl( _n , R ) obtained from the nth real square matrix ring Mn ( R ) by the difference of matrix products XY - YX = [ X , Y ].

For every Lie algebra ɡ, there exists a Lie group G such that L ( G )=ɡ.

The Lie algebra L ( H ) of a Lie subgroup H of a Lie group G is a Lie subalgebra of L ( G ), and conversely, for any Lie subalgebra η of L ( G ), there exists a unique Lie subgroup H of G such that L ( H )=η.

Furthermore, the differential df of a homomorphism f from _the Lie group G1 to _the Lie group G2 is a Lie algebra homomorphism from L ( G1 ₎ to L ( G2 ) _, and f is determined by df .

The above mainly discusses connected Lie groups, but Lie groups that are not connected manifolds can also be considered, and together with the theory of Lie algebras, Lie group theory is a complete branch of mathematics. In recent years, manifolds have also been considered over p -adic number fields, and the theory of p- adic Lie groups has been developed and is applied to number theory and other fields.

[Tsuneo Kanno]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

連結解析的多様体Gが同時に群であって、群の演算(g₁,g₂)→g₁g₂が積多様体G×GからGへの解析的写像で、さらに逆元をとる演算g→g^-1がGからGへの解析的写像であるとき、Gをリー群または解析的群という。

　このようにリー群は多様体と群の結合概念であるが、ノルウェーの数学者M・S・リーによって、幾何学や微分方程式を群論を用いて研究するため考え出されたもので、その名をとってリー群とよばれる。

　リー群Gは、実多様体か複素多様体かによって、それぞれ、実リー群、複素リー群ともいう。実数全体の加法群R、実一般線形群GL(n,R)は実リー群であり、複素数全体の加法群C、複素一般線形群GL(n,C)は複素リー群である。

　抽象的な群の場合と同じように、リー群の部分群、正規部分群、商群、準同型写像、同型写像などが、多様体としての自然な条件をあわせて考えて定義され、群論と類似のリー群論が完成されている。

　リー群論の特長は、リー群Gにリー代数ɡが、次に述べるように、自然に構成され、重要な役割を果たすところにある。

　リー群Gの元gに対し、解析的写像
　　Φ_g：G∋x→gx∈G
の微分dΦ_gは、Gの単位元eにおける接空間T(e)からgにおける接空間T(g)の上への線形空間の同型写像を与える。いま、X_e∈T(e)に対し、
　　X：G∋g→dΦ_g・(Xe)∈T(g)
で、多様体Gのベクトル場Xをつくる。このようなXの全体ɡ=｛X｜X_e∈T(e)｝はT(e)と同型な線形空間で、ベクトル場の積
　　〔X,Y〕=X・Y―Y・X
によって、一つの代数系となる。この代数ɡをL(G)と書き、リー群Gのリー代数という。