Affine root system

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 $u,v\; \backslash in\; E$, 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 $(\backslash cdot,\backslash cdot)$ on V.

This defines a metric on E as $d(u,v)=\backslash vert(u-v,u-v)\backslash vert$.

Consider the vector space F of affine-linear functions $f\backslash colon\; E\backslash longrightarrow\; \backslash mathbb\{R\}$.

Having fixed a $x\_0\backslash in\; E$, 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 $f^\backslash vee\; =\backslash frac\{2f\}\{(f,f)\}$ and $v^\backslash vee\; =\backslash frac\{2v\}\{(v,v)\}$ for any $f\backslash in\; F$ and $v\backslash in\; V$ respectively.

The identification let us define a reflection $w\_f$ over E in the following way:

:$w\_f(x)=x-f^\backslash vee(x)Df$

By transposition $w\_f$ acts also on F as

:$w\_f(g)=g-(f^\backslash vee,g)f$

An affine root system is a subset $S\backslash in\; F$ such that:

{{ordered list|start=1

|S spans F and its elements are non-constant.

|$w\_a(S)=S$ for every $a\backslash in\; S$.

|$(a,b^\backslash vee)\backslash in\backslash mathbb\{Z\}$ for every $a,b\backslash in\; S$.

}}

The elements of S are called affine roots.

Denote with $w(S)$ the group generated by the $w\_a$ with $a\backslash in\; S$.

We also ask

{{ordered list|start=4

|$w(S)$ as a discrete group acts properly on E.}}

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

The affine roots systems A₁ = B₁ = B{{su|b=1|p=∨}} = C₁ = C{{su|b=1|p=∨}} are the same, as are the pairs B₂ = C₂, B{{su|b=2|p=∨}} = C{{su|b=2|p=∨}}, 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.

:Rank 1: A₁, BC₁, (BC₁, C₁), (C{{su|b=1|p=∨}}, BC₁), (C{{su|b=1|p=∨}}, C₁).

:Rank 2: A₂, C₂, C{{su|b=2|p=∨}}, BC₂, (BC₂, C₂), (C{{su|b=2|p=∨}}, BC₂), (B₂, B{{su|b=2|p=∨}}), (C{{su|b=2|p=∨}}, C₂), G₂, G{{su|b=2|p=∨}}.

:Rank 3: A₃, B₃, B{{su|b=3|p=∨}}, C₃, C{{su|b=3|p=∨}}, BC₃, (BC₃, C₃), (C{{su|b=3|p=∨}}, BC₃), (B₃, B{{su|b=3|p=∨}}), (C{{su|b=3|p=∨}}, C₃).

:Rank 4: A₄, B₄, B{{su|b=4|p=∨}}, C₄, C{{su|b=4|p=∨}}, BC₄, (BC₄, C₄), (C{{su|b=4|p=∨}}, BC₄), (B₄, B{{su|b=4|p=∨}}), (C{{su|b=4|p=∨}}, C₄), D₄, F₄, F{{su|b=4|p=∨}}.

:Rank 5: A₅, B₅, B{{su|b=5|p=∨}}, C₅, C{{su|b=5|p=∨}}, BC₅, (BC₅, C₅), (C{{su|b=5|p=∨}}, BC₅), (B₅, B{{su|b=5|p=∨}}), (C{{su|b=5|p=∨}}, C₅), D₅.

:Rank 6: A₆, B₆, B{{su|b=6|p=∨}}, C₆, C{{su|b=6|p=∨}}, BC₆, (BC₆, C₆), (C{{su|b=6|p=∨}}, BC₆), (B₆, B{{su|b=6|p=∨}}), (C{{su|b=6|p=∨}}, C₆), D₆, E₆,

:Rank 7: A₇, B₇, B{{su|b=7|p=∨}}, C₇, C{{su|b=7|p=∨}}, BC₇, (BC₇, C₇), (C{{su|b=7|p=∨}}, BC₇), (B₇, B{{su|b=7|p=∨}}), (C{{su|b=7|p=∨}}, C₇), D₇, E₇,

:Rank 8: A₈, B₈, B{{su|b=8|p=∨}}, C₈, C{{su|b=8|p=∨}}, BC₈, (BC₈, C₈), (C{{su|b=8|p=∨}}, BC₈), (B₈, B{{su|b=8|p=∨}}), (C{{su|b=8|p=∨}}, C₈), D₈, E₈,

:Rank n (n>8): A_n, B_n, B{{su|b=n|p=∨}}, C_n, C{{su|b=n|p=∨}}, BC_n, (BC_n, C_n), (C{{su|b=n|p=∨}}, BC_n), (B_n, B{{su|b=n|p=∨}}), (C{{su|b=n|p=∨}}, C_n), D_n.

• {{harvtxt|Macdonald|1972}} showed that the affine root systems index Macdonald identities
• {{harvtxt|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.
• {{harvtxt|Macdonald|2003}} showed that affine roots systems index families of Macdonald polynomials.

• {{Citation | last1=Bruhat | first1=F. | last2=Tits | first2=Jacques | title=Groupes réductifs sur un corps local | url=http://www.numdam.org/item?id=PMIHES_1972__41__5_0 | mr=0327923 | year=1972 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | volume=41 | pages=5–251 | doi=10.1007/bf02715544| s2cid=125864274 | url-access=subscription }}
• {{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Affine root systems and Dedekind's η-function | doi=10.1007/BF01418931 | mr=0357528 | year=1972 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=15 | issue=2 | pages=91–143| bibcode=1971InMat..15...91M | s2cid=122115111 }}
• {{Citation | last=Macdonald | first=I. G. | title = Affine Hecke algebras and orthogonal polynomials | location=Cambridge | series=Cambridge Tracts in Mathematics | volume=157 | publisher=Cambridge University Press | year=2003 | pages=x+175 | isbn=978-0-521-82472-9| mr=1976581}}

Category:Discrete groups

Category:Lie algebras

Category:Orthogonal polynomials