Affine root system

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 Template:Harvtxt and Template:Harvtxt (except that both these papers accidentally omitted the Dynkin diagram Template:Dynkin).

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

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

Consider the vector space F of affine-linear functions $f:E⟶ℝ$. Having fixed a $x_0\in E$, every element in F can be written as $f(x)=Df(x-x_0)+f(x_0)$ with $Df$ a linear function on 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 $(f,g)=(Df,Dg)$. Set $f^{\vee }=\frac{2f}{(f,f)}$ and $v^{\vee }=\frac{2v}{(v,v)}$ for any $f\in F$ and $v\in V$ respectively. The identification let us define a reflection $w_f$ over E in the following way:

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

By transposition $w_f$ acts also on F as

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

An affine root system is a subset $S\in F$ such that: Template:Ordered list The elements of S are called affine roots. Denote with $w(S)$ the group generated by the $w_a$ with $a\in S$. We also ask Template:Ordered list This means that for any two compacts $K,H\subseteq E$ the elements of $w(S)$ such that $w(K)\cap H\ne \varnothing$ are a finite number.

Classification

The affine roots systems A₁ = B₁ = BTemplate:Su = C₁ = CTemplate:Su are the same, as are the pairs B₂ = C₂, BTemplate:Su = CTemplate:Su, 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	Template:Dynkin, Template:Dynkin2, Template:Dynkin2, Template:Dynkin2, ...
Bn (n ≥ 3)	2	Template:Dynkin, Template:Dynkin,Template:Dynkin, ...
BTemplate:Su (n ≥ 3)	2	Template:Dynkin, Template:Dynkin,Template:Dynkin, ...
Cn (n ≥ 2)	3	Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
CTemplate:Su (n ≥ 2)	3	Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
BCn (n ≥ 1)	2 if n=1, 3 if n ≥ 2	Template:Dynkin, Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
Dn (n ≥ 4)	1	Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
E6	1	Template:Dynkin
E7	1	Template:Dynkin2
E8	1	Template:Dynkin2
F4	2	Template:Dynkin
FTemplate:Su	2	Template:Dynkin
G2	2	Template:Dynkin
GTemplate:Su	2	Template:Dynkin
(BCn, Cn) (n ≥ 1)	3 if n=1, 4 if n≥2	Template:Dynkin, Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
(CTemplate:Su, BCn) (n ≥ 1)	3 if n=1, 4 if n≥2	Template:Dynkin, Template:Dynkin, Template:Dynkin, Template:Dynkin, ...
(Bn, BTemplate:Su) (n ≥ 2)	4 if n=2, 3 if n≥3	Template:Dynkin, Template:Dynkin, Template:Dynkin,Template:Dynkin, ...
(CTemplate:Su, Cn) (n ≥ 1)	4 if n=1, 5 if n≥2	Template:Dynkin, Template:Dynkin, Template:Dynkin, Template:Dynkin, ...

Irreducible affine root systems by rank

Rank 1: A₁, BC₁, (BC₁, C₁), (CTemplate:Su, BC₁), (CTemplate:Su, C₁).

Rank 2: A₂, C₂, CTemplate:Su, BC₂, (BC₂, C₂), (CTemplate:Su, BC₂), (B₂, BTemplate:Su), (CTemplate:Su, C₂), G₂, GTemplate:Su.

Rank 3: A₃, B₃, BTemplate:Su, C₃, CTemplate:Su, BC₃, (BC₃, C₃), (CTemplate:Su, BC₃), (B₃, BTemplate:Su), (CTemplate:Su, C₃).

Rank 4: A₄, B₄, BTemplate:Su, C₄, CTemplate:Su, BC₄, (BC₄, C₄), (CTemplate:Su, BC₄), (B₄, BTemplate:Su), (CTemplate:Su, C₄), D₄, F₄, FTemplate:Su.

Rank 5: A₅, B₅, BTemplate:Su, C₅, CTemplate:Su, BC₅, (BC₅, C₅), (CTemplate:Su, BC₅), (B₅, BTemplate:Su), (CTemplate:Su, C₅), D₅.

Rank 6: A₆, B₆, BTemplate:Su, C₆, CTemplate:Su, BC₆, (BC₆, C₆), (CTemplate:Su, BC₆), (B₆, BTemplate:Su), (CTemplate:Su, C₆), D₆, E₆,

Rank 7: A₇, B₇, BTemplate:Su, C₇, CTemplate:Su, BC₇, (BC₇, C₇), (CTemplate:Su, BC₇), (B₇, BTemplate:Su), (CTemplate:Su, C₇), D₇, E₇,

Rank 8: A₈, B₈, BTemplate:Su, C₈, CTemplate:Su, BC₈, (BC₈, C₈), (CTemplate:Su, BC₈), (B₈, BTemplate:Su), (CTemplate:Su, C₈), D₈, E₈,

Rank n (n>8): A_n, B_n, BTemplate:Su, C_n, CTemplate:Su, BC_n, (BC_n, C_n), (CTemplate:Su, BC_n), (B_n, BTemplate:Su), (CTemplate:Su, C_n), D_n.

Applications

• Template:Harvtxt showed that the affine root systems index Macdonald identities
• Template:Harvtxt 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.
• Template:Harvtxt showed that affine roots systems index families of Macdonald polynomials.

References

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