Lie群、Lie代数

Lie群，我们基本上就是指正交群，酉群，辛群以及一些例外群。
小维数的单Lie群（Simple Lie groups of small dimension）
同一行的群都有相同的Lie代数。在1维情形，群是可交换的和非单的。
The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
Dimension[注：弱抽象为相应维数的流形][注：对应的单Lie代数]单Lie代数对应的单Lie群，不同构者以逗号分隔开
1[R,S^1]一维非紧致单连通可交换单Lie群R, 一维紧致非单连通可交换单Lie群S^1=U(1)=SO(2)=Spin(2)
 一维可交换Lie群——用矩阵（线性变换）的形式来表示绕固定轴的旋转群SO(2)：
{{x'},{y'}}={{cosθ,-sinθ},{sinθ,cosθ}}{{x},{y}}={{xcosθ-ysinθ},{xcosθ-ysinθ}}
O(1,1)群还包含反射（如x->-x,y->y）相应行列式为-1。是一个连续的、单参数非紧致一维Lie群
O(1,1)使x^2-y^2不变，可用一个参数a表示群元{{cosha,sinha},{sinha,cosha}},a∈R（参数空间无界）,detM=1,是群的一个二维表示。
3[S^3,RP^3][A_1]三维紧致单连通单Lie群S^3=Sp(1)=SU(2)=Spin(3),三维紧致非单连通不可交换单Lie群SO(3)=PSU(2)(Compact)
一个三维不可交换Lie群——三维转动群SO(3)也称为特殊正交群.三维空间绕固定点的一个转动，习惯用Euler角（θ，Φ，Ψ）来描述一个转动。不是单连通的流形。
g^Ψ_z表示绕z轴转Ψ角
g^θ_x表示绕x轴转θ角
g^Φ_y表示绕y轴转Φ角
于是g=g^Φ_yg^θ_xg^Ψ_z
令θ，Φ=0，即绕固定点O和固定轴z轴旋转Ψ
则{{x'},{y'}}={{cosΨ,-sinΨ},{sinΨ,cosΨ}}{{x},{y}}={{xcosΨ-ysinΨ},{xcosΨ-ysinΨ}}
{{x'},{y'},{z'}}={{cosΨ,-sinΨ,0},{sinΨ,cosΨ,0},{0,0,1}}{{x},{y},{z}}={{xcosΨ-ysinΨ},{xcosΨ-ysinΨ},{z}}
仅满足g^tg=gg^t[而不要求det g>0]的线性变换所构成的群称为O(3)——正交群。
三维转动群SO(3)是3维连通紧致单线性Lie群，相应的实Lie代数so(3)是3维1秩紧致实单Lie代数。
二维Lorentz群SO(2,1)是3维非紧致单线性Lie群，相应的实Lie代数so(2,1)是3维1秩非紧致实单Lie代数。
3 SL(2,R)=Sp(2,R),三维非紧致单Lie群SO(2,1)
O(2,1)是一个连续的、三参数非紧致三维Lie群，对易关系是：[I_1,I_2]=iI_3,[I_2,I_3]=-iI_1,[I_1,I_3]=-iI_2,
6 SL(2,C)=Sp(2,C), SO(3,1), SO(3,C)
8 SL(3,R)
8 SU(3)
8 SU(1,2)
10 Sp(2)=Spin(5), SO(5)
10 SO(4,1), Sp(2,2)
10 SO(3,2),Sp(4,R)
14 G2 (Compact)
14 G2 (Split)
15 SU(4)=Spin(6), SO(6)
15 SL(4,R), SO(3,3)
15 SU(3,1)
15 SU(2,2), SO(4,2)
15 SL(2,H), SO(5,1)
16 SL(3,C)
20 SO(5,C), Sp(4,C)
从1883年起，索非斯·李（Sophus Lie,1842.12.17-1899.2.18）等人开始研究Lie代数的结构，而且得出四个类型局部单Lie群，即射影线性群，射影正交群及射影辛群，这就是后来的典型Lie群(Lie代数)的来源。
域F上一个Lie代数g是所谓单的，即指除了g本身和{0}以外,g不含其他理想[构成Lie代数的环是单的就叫单Lie代数]。
域F上一个有限维Lie代数g是所谓半单的，即指g不含非零可解理想。
每一个有限维Lie代数g都含有惟一的最大可解理想r,就是这样一个理想, 它包含g的一切可解理想，称为g的根基。
g是半单的当且仅当它的根基r＝{0}。
特征为0的域上每一个半单Lie代数都是一些单Lie代数的直和。
定理：sl(l+1,C)(l>=1)是单Lie代数，其Dynkin图为A_l。
gap> for n in [1..5] do An:=SimpleLieAlgebra("A",n,Rationals);Print(An,"\n");od;
Algebra( Rationals, [ v.1, v.2, v.3 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35 ] )
SO(3)有三个单参数子群，g_x(t)，g_y(t)，g_z(t)，它的Lie代数是A_1，元素可表示为(a_1,a_2,a_3)={{0,-a_3,a_2},{a_3,0,-a_1},{-a_2,a_1,0}},[a,b]=a×b。
SO(4)有六个单参数子群，g_zh(t)，g_yh(t)，g_xh(t)，g_xy(t)，g_xz(t)，g_yz(t)，它的Lie代数是D_2，元素可表示为
(a_1,a_2,a_3,a_4,a_5,a_6)={{0,-a_1,a_2,-a_6},{a_1,0,-a_3,-a_5},{-a_2,a_3,0,-a_4},{a_6,a_5,a_4,0}},[a,b]=(-[123]_1-[456]_1,-[123]_2-[456]_2,-[123]_3-[456]_3,-[126]_1-[345]_2,-[126]_2-[345]_3,-[156]_3-[234]_2)，其中，记[ijk]=([ijk]_1,[ijk]_2,[ijk]_3)=(a_i,a_j,a_k)×(b_i,b_j,b_k)。
{(a_1,a_2,a_3,0,0,0)}构成D_2的子Lie代数A_1，但{(0,0,0,a_4,a_5,a_6)}不构成D_2的子Lie代数。
{(a_1,0,0,0,a_5,a_6)}构成D_2的子Lie代数A_1，但{(0,a_2,a_3,a_4,0,0)}不构成D_2的子Lie代数。
定理：so(2l+1,C)(l>=1)是单Lie代数，其Dynkin图为B_l。
gap> for n in [2..5] do Bn:=SimpleLieAlgebra("B",n,Rationals);Print(Bn,"\n");od;
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35, v.36 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35, v.36,
  v.37, v.38, v.39, v.40, v.41, v.42, v.43, v.44, v.45, v.46, v.47, v.48, v.49, v.50, v.51, v.52, v.53, v.54, v.55 ] )
定理：sp(l,C)是单Lie代数，其Dynkin图为C_l。
gap> for n in [2..5] do Cn:=SimpleLieAlgebra("C",n,Rationals);Print(Cn,"\n");od;
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35, v.36 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35, v.36,
  v.37, v.38, v.39, v.40, v.41, v.42, v.43, v.44, v.45, v.46, v.47, v.48, v.49, v.50, v.51, v.52, v.53, v.54, v.55 ] )
定理：so(2l,C)(l>=4)是单Lie代数，其Dynkin图为D_l。
gap> for n in [4..5] do Dn:=SimpleLieAlgebra("D",n,Rationals);Print(Dn,"\n");od;
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28 ] )
Algebra( Rationals, [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.15, v.16, v.17,
  v.18, v.19, v.20, v.21, v.22, v.23, v.24, v.25, v.26, v.27, v.28, v.29, v.30, v.31, v.32, v.33, v.34, v.35, v.36,
  v.37, v.38, v.39, v.40, v.41, v.42, v.43, v.44, v.45 ] )
1888-1890年，德国数学家基林(W.Killing，1847-1923)更找出例外的单Lie群。
gap> G2:=SimpleLieAlgebra( "G", 2, Rationals );SemiSimpleType(G2);B:=KillingMatrix(Basis(G2));;G2C:=LieCenter(G2);DG2:=LieDerivedSubalgebra(G2);IsSimpleAlgebra(G2);IsFiniteDimensional(G2);IsAbelian(G2);IsLieAbelian(G2);IsLieSolvable(G2);
<Lie algebra of dimension 14 over Rationals>
"G2"
<Lie algebra of dimension 0 over Rationals>
<Lie algebra of dimension 14 over Rationals>
true
true
false
false
false
gap> F4:=SimpleLieAlgebra( "F", 4, Rationals );SemiSimpleType(F4);B:=KillingMatrix(Basis(F4));;F4C:=LieCenter(F4);DF4:=LieDerivedSubalgebra(F4);IsSimpleAlgebra(F4);IsFiniteDimensional(F4);IsAbelian(F4);IsLieAbelian(F4);IsLieSolvable(F4);
<Lie algebra of dimension 52 over Rationals>
"F4"
<Lie algebra of dimension 0 over Rationals>
<Lie algebra of dimension 52 over Rationals>
true
true
false
false
false
gap> E6:=SimpleLieAlgebra( "E", 6, Rationals );SemiSimpleType(E6);B:=KillingMatrix(Basis(E6));;E6C:=LieCenter(E6);DE6:=LieDerivedSubalgebra(E6);IsSimpleAlgebra(E6);IsFiniteDimensional(E6);IsAbelian(E6);IsLieAbelian(E6);IsLieSolvable(E6);
<Lie algebra of dimension 78 over Rationals>
"E6"
<Lie algebra of dimension 0 over Rationals>
<Lie algebra of dimension 78 over Rationals>
true
true
false
false
false
gap> E7:=SimpleLieAlgebra( "E", 7, Rationals );SemiSimpleType(E7);B:=KillingMatrix(Basis(E7));;E7C:=LieCenter(E7);DE7:=LieDerivedSubalgebra(E7);IsSimpleAlgebra(E7);IsFiniteDimensional(E7);IsAbelian(E7);IsLieAbelian(E7);IsLieSolvable(E7);
<Lie algebra of dimension 133 over Rationals>
"E7"
<Lie algebra of dimension 0 over Rationals>
<Lie algebra of dimension 133 over Rationals>
true
true
false
false
false
gap> E8:=SimpleLieAlgebra( "E", 8, Rationals );SemiSimpleType(E8);B:=KillingMatrix(Basis(E8));;E8C:=LieCenter(E8);DE8:=LieDerivedSubalgebra(E8);IsSimpleAlgebra(E8);IsFiniteDimensional(E8);IsAbelian(E8);IsLieAbelian(E8);IsLieSolvable(E8);
<Lie algebra of dimension 248 over Rationals>
"E8"
<Lie algebra of dimension 0 over Rationals>
<Lie algebra of dimension 248 over Rationals>
true
true
false
false
false