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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 ).

Definition

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 ${\displaystyle u,v\in E}$, then it is well defined an element in V denoted as ${\displaystyle u-v}$ which is the only element w such that ${\displaystyle v+w=u}$.

Now suppose we have a scalar product ${\displaystyle (\cdot ,\cdot )}$ on V. This defines a metric on E as ${\displaystyle d(u,v)=\vert (u-v,u-v)\vert }$.

Consider the vector space F of affine-linear functions ${\displaystyle f\colon E\longrightarrow \mathbb {R} }$. Having fixed a ${\displaystyle x_{0}\in E}$, every element in F can be written as ${\displaystyle f(x)=Df(x-x_{0})+f(x_{0})}$ with ${\displaystyle Df}$ a linear function on V that doesn't depend on the choice of ${\displaystyle x_{0}}$.

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 ${\displaystyle (f,g)=(Df,Dg)}$. Set ${\displaystyle f^{\vee }={\frac {2f}{(f,f)}}}$ and ${\displaystyle v^{\vee }={\frac {2v}{(v,v)}}}$ for any ${\displaystyle f\in F}$ and ${\displaystyle v\in V}$ respectively. The identification let us define a reflection ${\displaystyle w_{f}}$ over E in the following way:

${\displaystyle w_{f}(x)=x-f^{\vee }(x)Df}$

By transposition ${\displaystyle w_{f}}$ acts also on F as

${\displaystyle w_{f}(g)=g-(f^{\vee },g)f}$

An affine root system is a subset ${\displaystyle S\in F}$ such that:

1. S spans F and its elements are non-constant.
2. ${\displaystyle w_{a}(S)=S}$ for every ${\displaystyle a\in S}$.
3. ${\displaystyle (a,b^{\vee })\in \mathbb {Z} }$ for every ${\displaystyle a,b\in S}$.

The elements of S are called affine roots. Denote with ${\displaystyle w(S)}$ the group generated by the ${\displaystyle w_{a}}$ with ${\displaystyle a\in S}$. We also ask

4. ${\displaystyle w(S)}$ as a discrete group acts properly on E.

This means that for any two compacts ${\displaystyle K,H\subseteq E}$ the elements of ${\displaystyle w(S)}$ such that ${\displaystyle w(K)\cap H\neq \varnothing }$ are a finite number.

Classification

The affine roots systems A₁ = B₁ = B
₁ = C₁ = C
₁ are the same, as are the pairs B₂ = C₂, B
₂ = C
₂, and A₃ = D₃

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
An (n ≥ 1)	2 if n=1, 1 if n≥2	, , , , ...
Bn (n ≥ 3)	2	, ,, ...
B n (n ≥ 3)	2	, ,, ...
Cn (n ≥ 2)	3	, , , ...
C n (n ≥ 2)	3	, , , ...
BCn (n ≥ 1)	2 if n=1, 3 if n ≥ 2	, , , , ...
Dn (n ≥ 4)	1	, , , ...
E6	1
E7	1
E8	1
F4	2
F 4	2
G2	2
G 2	2
(BCn, Cn) (n ≥ 1)	3 if n=1, 4 if n≥2	, , , , ...
(C n, BCn) (n ≥ 1)	3 if n=1, 4 if n≥2	, , , , ...
(Bn, B n) (n ≥ 2)	4 if n=2, 3 if n≥3	, , ,, ...
(C n, Cn) (n ≥ 1)	4 if n=1, 5 if n≥2	, , , , ...

Irreducible affine root systems by rank

Rank 1: A₁, BC₁, (BC₁, C₁), (C
₁, BC₁), (C
₁, C₁).

Rank 2: A₂, C₂, C
₂, BC₂, (BC₂, C₂), (C
₂, BC₂), (B₂, B
₂), (C
₂, C₂), G₂, G
₂.

Rank 3: A₃, B₃, B
₃, C₃, C
₃, BC₃, (BC₃, C₃), (C
₃, BC₃), (B₃, B
₃), (C
₃, C₃).

Rank 4: A₄, B₄, B
₄, C₄, C
₄, BC₄, (BC₄, C₄), (C
₄, BC₄), (B₄, B
₄), (C
₄, C₄), D₄, F₄, F
₄.

Rank 5: A₅, B₅, B
₅, C₅, C
₅, BC₅, (BC₅, C₅), (C
₅, BC₅), (B₅, B
₅), (C
₅, C₅), D₅.

Rank 6: A₆, B₆, B
₆, C₆, C
₆, BC₆, (BC₆, C₆), (C
₆, BC₆), (B₆, B
₆), (C
₆, C₆), D₆, E₆,

Rank 7: A₇, B₇, B
₇, C₇, C
₇, BC₇, (BC₇, C₇), (C
₇, BC₇), (B₇, B
₇), (C
₇, C₇), D₇, E₇,

Rank 8: A₈, B₈, B
₈, C₈, C
₈, BC₈, (BC₈, C₈), (C
₈, BC₈), (B₈, B
₈), (C
₈, C₈), D₈, E₈,

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.

Applications

• Macdonald (1972) showed that the affine root systems index Macdonald identities
• Bruhat & Tits (1972) used affine root systems to study p-adic algebraic groups.
• Reduced affine root systems classify affine Kac–Moody algebras, while the non-reduced affine root systems correspond to affine Lie superalgebras.
• Macdonald (2003) showed that affine roots systems index families of Macdonald polynomials.

References

• Bruhat, F.; Tits, Jacques (1972), "Groupes réductifs sur un corps local", Publications Mathématiques de l'IHÉS, 41: 5–251, doi:10.1007/bf02715544, ISSN 1618-1913, MR 0327923, S2CID 125864274
• Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae, 15 (2): 91–143, Bibcode:1971InMat..15...91M, doi:10.1007/BF01418931, ISSN 0020-9910, MR 0357528, S2CID 122115111
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9, MR 1976581

• Discrete groups
• Lie algebras
• Orthogonal polynomials