File:G2 affine chamber.svg The affine root system of type G 2 . In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space . They are used in the classification of affine Lie algebras and superalgebras, and semisimple p -adic algebraic groups , and correspond to families of Macdonald polynomials . The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras . Possibly non-reduced affine root systems were introduced and classified by Macdonald (1972) and Bruhat & Tits (1972) (except that both these papers accidentally omitted the Dynkin diagram File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png
Let E be an affine space and V the vector space of its translations.
Recall that V acts faithfully and transitively on E .
In particular, if u , v ∈ E V denoted as u − v v + w = u
Now suppose we have a scalar product ( ⋅ , ⋅ ) V .
This defines a metric on E as d ( u , v ) = | ( u − v , u − v ) |
Consider the vector space F of affine-linear functions f : E ⟶ ℝ x 0 ∈ E F can be written as f ( x ) = D f ( x − x 0 ) + f ( x 0 ) D f V that doesn't depend on the choice of x 0
Now the dual of V can be identified with V thanks to the chosen scalar product and we can define a product on F as ( f , g ) = ( D f , D g ) f ∨ = 2 f ( f , f ) v ∨ = 2 v ( v , v ) f ∈ F v ∈ V w f E in the following way:
w f ( x ) = x − f ∨ ( x ) D f By transposition w f F as
w f ( g ) = g − ( f ∨ , g ) f An affine root system is a subset S ⊂ F
S spans F and its elements are non-constant. w a ( S ) = S a ∈ S ( a , b ∨ ) ∈ ℤ a , b ∈ S The elements of S are called affine roots .
Denote with w ( S ) w a a ∈ S
w ( S ) E .This means that for any two compacts K , H ⊆ E w ( S ) w ( K ) ∩ H ≠ ∅
The affine roots systems A 1 = B 1 = B ∨ 1 C 1 = C ∨ 1 B 2 = C 2 , B ∨ 2 C ∨ 2 A 3 = D 3
The number of orbits given in the table is the number of orbits of simple roots under the Weyl group.
In the Dynkin diagrams , the non-reduced simple roots α (with 2α a root) are colored green. The first Dynkin diagram in a series sometimes does not follow the same rule as the others.
Affine root system
Number of orbits
Dynkin diagram
A n n ≥ 1)2 if n =1, 1 if n ≥2
File:Dyn-node.png File:Dyn-4ab.png File:Dyn-node.png File:Dyn2-branch.png File:Dyn2-loop2.png File:Dyn2-loop1.png File:Dyn2-nodes.png File:Dyn2-loop2.png File:Dyn2-branch.png File:Dyn2-3s.png File:Dyn2-nodes.png File:Dyn2-loop2.png
B n n ≥ 3)2
File:Dyn-branch1.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png
B ∨ n n ≥ 3)2
File:Dyn-branch1.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png
C n n ≥ 2)3
File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png
C ∨ n n ≥ 2)3
File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png
BC n n ≥ 1)2 if n =1, 3 if n ≥ 2
File:Dyn-node.png File:Dyn-4c.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png
D n n ≥ 4)1
File:Dyn-branch1.png File:Dyn-node.png File:Dyn-branch2.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-branch2.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-branch2.png
E 6 1
File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-branch2.png File:Dyn-3s.png File:Dyn-nodes.png
E 7 1
File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-branch.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png
E 8 1
File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-branch.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png File:Dyn2-3.png File:Dyn2-node.png
F 4 2
File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png
F ∨ 4 2
File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png
G 2 2
File:Dyn-node.png File:Dyn-6a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png
G ∨ 2 2
File:Dyn-node.png File:Dyn-6b.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png
(BC n C n n ≥ 1)
3 if n =1, 4 if n ≥2
File:Dyn-nodeg.png File:Dyn-4c.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4a.png File:Dyn-node.png
(C ∨ n BC n n ≥ 1)
3 if n =1, 4 if n ≥2
File:Dyn-nodeg.png File:Dyn-4ab.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-node.png
(B n B ∨ n n ≥ 2)
4 if n =2, 3 if n ≥3
File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-branch1.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png
(C ∨ n C n n ≥ 1)
4 if n =1, 5 if n ≥2
File:Dyn-nodeg.png File:Dyn-4ab.png File:Dyn-nodeg.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png File:Dyn-nodeg.png File:Dyn-4a.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-3.png File:Dyn-node.png File:Dyn-4b.png File:Dyn-nodeg.png
Rank 1 : A 1 , BC 1 , (BC 1 , C 1 ), (C ∨ 1 BC 1 ), (C ∨ 1 C 1 ).Rank 2 : A 2 , C 2 , C ∨ 2 BC 2 , (BC 2 , C 2 ), (C ∨ 2 BC 2 ), (B 2 , B ∨ 2 C ∨ 2 C 2 ), G 2 , G ∨ 2 Rank 3 : A 3 , B 3 , B ∨ 3 C 3 , C ∨ 3 BC 3 , (BC 3 , C 3 ), (C ∨ 3 BC 3 ), (B 3 , B ∨ 3 C ∨ 3 C 3 ).Rank 4 : A 4 , B 4 , B ∨ 4 C 4 , C ∨ 4 BC 4 , (BC 4 , C 4 ), (C ∨ 4 BC 4 ), (B 4 , B ∨ 4 C ∨ 4 C 4 ), D 4 , F 4 , F ∨ 4 Rank 5 : A 5 , B 5 , B ∨ 5 C 5 , C ∨ 5 BC 5 , (BC 5 , C 5 ), (C ∨ 5 BC 5 ), (B 5 , B ∨ 5 C ∨ 5 C 5 ), D 5 .Rank 6 : A 6 , B 6 , B ∨ 6 C 6 , C ∨ 6 BC 6 , (BC 6 , C 6 ), (C ∨ 6 BC 6 ), (B 6 , B ∨ 6 C ∨ 6 C 6 ), D 6 , E 6 ,Rank 7 : A 7 , B 7 , B ∨ 7 C 7 , C ∨ 7 BC 7 , (BC 7 , C 7 ), (C ∨ 7 BC 7 ), (B 7 , B ∨ 7 C ∨ 7 C 7 ), D 7 , E 7 ,Rank 8 : A 8 , B 8 , B ∨ 8 C 8 , C ∨ 8 BC 8 , (BC 8 , C 8 ), (C ∨ 8 BC 8 ), (B 8 , B ∨ 8 C ∨ 8 C 8 ), D 8 , E 8 ,Rank n (n >8) : A n B n B ∨ n C n C ∨ n BC n BC n C n C ∨ n BC n B n B ∨ n C ∨ n C n D n Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). 
