The affine root system of type G₂.

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:

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

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	, , , , ...

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