設
V
{\displaystyle V}
向量空間 ，並賦予標準的內積
(
,
)
{\displaystyle (,)}
V
{\displaystyle V}
根系 是有限個向量（稱為根 ）構成的集合
Φ
{\displaystyle \Phi }
<α, β> 的整性條件使得 β 必然落在所示各條垂直線上。再配合 <β, α> 的整性條件，在每條線上，其間交角只有兩種可能。
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
對任一
α
∈
Φ
{\displaystyle \alpha \in \Phi }
Φ
{\displaystyle \Phi }
±
α
{\displaystyle \pm \alpha }
對任意
α
∈
Φ
{\displaystyle \alpha \in \Phi }
Φ
{\displaystyle \Phi }
α
{\displaystyle \alpha }
σ
α
(
β
)
=
β
−
2
(
α
,
β
)
(
α
,
α
)
α
∈
Φ
.
{\displaystyle \sigma _{\alpha }(\beta )=\beta -2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\alpha \in \Phi .}
（整性）若
α
,
β
∈
Φ
{\displaystyle \alpha ,\beta \in \Phi }
β
{\displaystyle \beta }
α
{\displaystyle \alpha }
α
{\displaystyle \alpha }
⟨
β
,
α
⟩
:=
2
(
α
,
β
)
(
α
,
α
)
∈
Z
,
{\displaystyle \langle \beta ,\alpha \rangle :=2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\in \mathbb {Z} ,}
根據性質三，整性等價於：對任意
α
,
β
∈
Φ
{\displaystyle \alpha ,\beta \in \Phi }
σ
α
(
β
)
{\displaystyle \sigma _{\alpha }(\beta )}
β
{\displaystyle \beta }
α
{\displaystyle \alpha }
尖積
⟨
⋅
,
⋅
⟩
:
Φ
×
Φ
→
Z
{\displaystyle \langle \cdot ,\cdot \rangle \colon \Phi \times \Phi \to \mathbb {Z} }
並非一個內積，它未必對稱，而且只對第一個參數是線性的。
根系
Φ
{\displaystyle \Phi }
秩 定義為
V
{\displaystyle V}
給定兩個根系
(
V
,
Φ
)
,
(
W
,
Ψ
)
{\displaystyle (V,\Phi ),(W,\Psi )}
V
⊕
W
{\displaystyle V\oplus W}
Φ
⊔
Ψ
{\displaystyle \Phi \sqcup \Psi }
V
,
W
≠
{
0
}
{\displaystyle V,W\neq \{0\}}
不可約 的。
對兩個根系
(
E
1
,
Φ
1
)
,
(
E
2
,
Φ
2
)
{\displaystyle (E_{1},\Phi _{1}),(E_{2},\Phi _{2})}
Φ
1
{\displaystyle \Phi _{1}}
Φ
2
{\displaystyle \Phi _{2}}
對於根系
(
V
,
Φ
)
{\displaystyle (V,\Phi )}
外爾群 。可證明此群在
Φ
{\displaystyle \Phi }
秩为1的例子 编辑 在同構的意義下，秩一的根系僅有一種，由兩個非零向量
{
α
,
−
α
}
{\displaystyle \{\alpha ,-\alpha \}}
A
1
{\displaystyle A_{1}}
秩为2的例子 编辑 秩二的根系有四種可能，对应于
σ
α
(
β
)
=
β
+
n
α
{\displaystyle \sigma _{\alpha }(\beta )=\beta +n\alpha }
n
=
0
,
1
,
2
,
3
{\displaystyle n=0,1,2,3}
[1]
A
1
×
A
1
{\displaystyle A_{1}\times A_{1}}
B
2
{\displaystyle B_{2}}
正方形格 ，而
A
2
{\displaystyle A_{2}}
G
2
{\displaystyle G_{2}}
六边形格 。这仅仅是五种可能的二维格 中的两种。
圖解如下：
根系 A1 ×A1
根系 A2 根系 B2
根系 G2
秩二之根系
當
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
W
{\displaystyle W}
Ψ
=
Φ
∩
W
{\displaystyle \Psi =\Phi \cap W}
W
{\displaystyle W}
Ψ
{\displaystyle \Psi }
W
{\displaystyle W}
0
,
30
,
45
,
60
,
90
,
120
,
135
,
150
{\displaystyle 0,30,45,60,90,120,135,150}
180
{\displaystyle 180}
取
V
{\displaystyle V}
R
n
+
1
{\displaystyle \mathbb {R} ^{n+1}}
∑
i
=
1
n
x
i
=
0
{\displaystyle \sum _{i=1}^{n}x_{i}=0}
(
x
1
,
…
,
x
n
)
{\displaystyle (x_{1},\ldots ,x_{n})}
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
2
{\displaystyle {\sqrt {2}}}
R
n
+
1
{\displaystyle \mathbb {R} ^{n+1}}
e
1
,
…
,
e
n
+
1
{\displaystyle e_{1},\ldots ,e_{n+1}}
e
i
−
e
j
(
i
≠
j
)
{\displaystyle e_{i}-e_{j}\;(i\neq j)}
n
(
n
+
1
)
{\displaystyle n(n+1)}
α
i
:=
e
i
−
e
i
+
1
{\displaystyle \alpha _{i}:=e_{i}-e_{i+1}}
對垂直於
α
i
{\displaystyle \alpha _{i}}
超平面 的鏡射在
Φ
{\displaystyle \Phi }
i
,
i
+
1
{\displaystyle i,i+1}
A
n
{\displaystyle A_{n}}
S
n
+
1
{\displaystyle S_{n+1}}
A
n
{\displaystyle A_{n}}
s
l
(
n
+
1
,
C
)
{\displaystyle {\mathfrak {sl}}(n+1,\mathbb {C} )}
B 4 1
-1
0
0
0
1
-1
0
0
0
1
-1
0
0
0
1
取
V
=
R
n
{\displaystyle V=\mathbb {R} ^{n}}
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
1
,
2
{\displaystyle 1,{\sqrt {2}}}
2
n
2
{\displaystyle 2n^{2}}
α
i
=
e
i
−
e
i
+
1
(
1
≤
i
<
n
)
{\displaystyle \alpha _{i}=e_{i}-e_{i+1}\;(1\leq i<n)}
α
n
:=
e
n
{\displaystyle \alpha _{n}:=e_{n}}
對短根
α
n
{\displaystyle \alpha _{n}}
(
x
1
,
…
,
x
n
)
↦
(
x
1
,
…
,
−
x
n
)
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto (x_{1},\ldots ,-x_{n})}
B
1
{\displaystyle B_{1}}
A
1
{\displaystyle A_{1}}
n
≥
2
{\displaystyle n\geq 2}
B
n
{\displaystyle B_{n}}
s
o
(
2
n
+
1
,
C
)
{\displaystyle {\mathfrak {so}}(2n+1,\mathbb {C} )}
C 4 1
-1
0
0
0
1
-1
0
0
0
1
-1
0
0
0
2
取
V
=
R
n
{\displaystyle V=\mathbb {R} ^{n}}
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
2
{\displaystyle {\sqrt {2}}}
2
λ
{\displaystyle 2\lambda }
λ
{\displaystyle \lambda }
2
n
2
{\displaystyle 2n^{2}}
α
i
:=
e
i
−
e
i
+
1
(
1
≤
i
<
n
)
{\displaystyle \alpha _{i}:=e_{i}-e_{i+1}\;(1\leq i<n)}
α
n
:=
2
e
n
{\displaystyle \alpha _{n}:=2e_{n}}
C
2
{\displaystyle C_{2}}
B
2
{\displaystyle B_{2}}
n
≥
3
{\displaystyle n\geq 3}
C
n
{\displaystyle C_{n}}
s
p
(
2
n
,
C
)
{\displaystyle {\mathfrak {sp}}(2n,\mathbb {C} )}
D 4 1
-1
0
0
0
1
-1
0
0
0
1
-1
0
0
1
1
取
V
:=
R
n
{\displaystyle V:=\mathbb {R} ^{n}}
Φ
{\displaystyle \Phi }
V
{\displaystyle V}
2
{\displaystyle {\sqrt {2}}}
2
n
(
n
−
1
)
{\displaystyle 2n(n-1)}
α
i
=
e
i
−
e
i
+
1
,
(
1
≤
i
<
n
)
{\displaystyle \alpha _{i}=e_{i}-e_{i+1},\;(1\leq i<n)}
α
n
=
e
n
+
e
n
−
1
{\displaystyle \alpha _{n}=e_{n}+e_{n-1}}
D
3
{\displaystyle D_{3}}
A
3
{\displaystyle A_{3}}
n
≥
4
{\displaystyle n\geq 4}
D
n
{\displaystyle D_{n}}
s
o
(
2
n
,
C
)
{\displaystyle {\mathfrak {so}}(2n,\mathbb {C} )}
E8 , E7 , E6 编辑
E
8
{\displaystyle E_{8}}
R
8
{\displaystyle \mathbb {R} ^{8}}
Γ
8
{\displaystyle \Gamma _{8}}
各座標均為整數，或均為半整數（不容相混）。
八個座標的和為偶數。 定義
E
8
{\displaystyle E_{8}}
Γ
8
{\displaystyle \Gamma _{8}}
2
{\displaystyle {\sqrt {2}}}
{
α
∈
Z
8
⊔
(
Z
+
1
2
)
8
:
|
α
|
2
=
2
,
∑
α
i
∈
2
Z
}
{\displaystyle \{\alpha \in \mathbb {Z} ^{8}\sqcup (\mathbb {Z} +{\frac {1}{2}})^{8}:|\alpha |^{2}=2,\;\sum \alpha _{i}\in 2\mathbb {Z} \}}
定義
E
7
{\displaystyle E_{7}}
E
8
{\displaystyle E_{8}}
{
x
:
(
x
,
α
)
=
0
}
{\displaystyle \{x:(x,\alpha )=0\}}
α
∈
E
8
{\displaystyle \alpha \in E_{8}}
E
7
{\displaystyle E_{7}}
E
6
{\displaystyle E_{6}}
E
6
,
E
7
,
E
8
{\displaystyle E_{6},E_{7},E_{8}}
D
5
,
A
4
{\displaystyle D_{5},A_{4}}
E 8 1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
1
1
0
½
½
½
½
½
½
½
½
另一種等價的描述是取
Γ
8
′
{\displaystyle \Gamma '_{8}}
各坐標均為整數，而且其和為偶數；或
各坐標均為半整數，而且其和為奇數。
Γ
8
{\displaystyle \Gamma _{8}}
Γ
8
′
{\displaystyle \Gamma '_{8}}
Γ
8
{\displaystyle \Gamma _{8}}
E
8
{\displaystyle E_{8}}
Γ
8
′
{\displaystyle \Gamma '_{8}}
在偶坐標下，通常取單根為
α
i
:=
e
i
−
e
i
+
1
(
1
≤
i
≤
6
)
{\displaystyle \alpha _{i}:=e_{i}-e_{i+1}\quad (1\leq i\leq 6)}
α
7
:=
e
7
+
e
6
{\displaystyle \alpha _{7}:=e_{7}+e_{6}}
α
8
=
β
0
=
∑
i
=
1
8
e
i
2
{\displaystyle \alpha _{8}=\beta _{0}={\frac {\sum _{i=1}^{8}e_{i}}{2}}}
E 8 1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
1
-1
-½
-½
-½
-½
-½
½
½
½
在奇坐標下，通常取單根為
α
i
:=
e
i
−
e
i
+
1
(
1
≤
i
≤
7
)
{\displaystyle \alpha _{i}:=e_{i}-e_{i+1}\quad (1\leq i\leq 7)}
α
8
:=
β
5
{\displaystyle \alpha _{8}:=\beta _{5}}
β
j
:=
−
∑
i
=
1
j
e
i
+
∑
i
=
j
+
1
8
e
i
2
{\displaystyle \beta _{j}:={\frac {-\sum _{i=1}^{j}e_{i}+\sum _{i=j+1}^{8}e_{i}}{2}}}
（在上述定義中，若改取
β
3
{\displaystyle \beta _{3}}
β
1
,
β
7
,
β
2
,
β
6
{\displaystyle \beta _{1},\beta _{7},\beta _{2},\beta _{6}}
A
8
{\displaystyle A_{8}}
D
8
{\displaystyle D_{8}}
β
4
{\displaystyle \beta _{4}}
α
1
,
…
,
α
7
{\displaystyle \alpha _{1},\ldots ,\alpha _{7}}
刪去
α
1
{\displaystyle \alpha _{1}}
E
7
{\displaystyle E_{7}}
α
2
{\displaystyle \alpha _{2}}
E
6
{\displaystyle E_{6}}
由於對
α
1
{\displaystyle \alpha _{1}}
α
1
,
α
2
{\displaystyle \alpha _{1},\alpha _{2}}
E
7
,
E
6
{\displaystyle E_{7},E_{6}}
E7 α ∈ Z 7 ∪ (Z +½)7 : ∑α i 2 + α 1 2 = 2，∑α i + α 1 ∈ 2Z ),
E6 α ∈ Z 6 ∪ (Z +½)6 : ∑α i 2 + 2α 1 2 = 2，∑α i + 2α 1 ∈ 2Z )
F 4 1
-1
0
0
0
1
-1
0
0
0
1
0
-½
-½
-½
-½
對於
F
4
{\displaystyle F_{4}}
V
=
R
4
{\displaystyle V=\mathbb {R} ^{4}}
Φ
{\displaystyle \Phi }
|
α
|
=
1
,
2
{\displaystyle |\alpha |=1,{\sqrt {2}}}
2
α
{\displaystyle 2\alpha }
此根系有
48
{\displaystyle 48}
B
3
{\displaystyle B_{3}}
α
4
=
−
(
∑
i
=
1
4
e
i
)
/
2
{\displaystyle \alpha _{4}=-(\sum _{i=1}^{4}e_{i})/2}
G
2
{\displaystyle G_{2}}
α
1
{\displaystyle \alpha _{1}}
β
:=
α
2
−
α
1
{\displaystyle \beta :=\alpha _{2}-\alpha _{1}}
在此沿用了之前的符號：
α
i
:=
e
i
−
e
i
+
1
,
(
i
=
1
,
2
)
{\displaystyle \alpha _{i}:=e_{i}-e_{i+1},\;(i=1,2)}
关于一棵植物的根的整體，请见「根 § 根系」。
