Affine root system

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 and (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 $u,v \in E$, 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 $d(u,v)=\vert(u-v,u-v)\vert$.

Consider the vector space ''F'' of affine-linear functions $f\colon E\longrightarrow \mathbb$.
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 $f^\vee =\frac$ and $v^\vee =\frac$ 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:

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

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

Classification

The affine roots systems ''A''₁ = ''B''₁ = ''B'' = ''C''₁ = ''C'' are the same, as are the pairs ''B''₂ = ''C''₂, ''B'' = ''C'', and ''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.

Irreducible affine root systems by rank

:Rank 1: ''A''₁, ''BC''₁, (''BC''₁, ''C''₁), (''C'', ''BC''₁), (''C'', ''C''₁).
:Rank 2: ''A''₂, ''C''₂, ''C'', ''BC''₂, (''BC''₂, ''C''₂), (''C'', ''BC''₂), (''B''₂, ''B''), (''C'', ''C''₂), ''G''₂, ''G''.
:Rank 3: ''A''₃, ''B''₃, ''B'', ''C''₃, ''C'', ''BC''₃, (''BC''₃, ''C''₃), (''C'', ''BC''₃), (''B''₃, ''B''), (''C'', ''C''₃).
:Rank 4: ''A''₄, ''B''₄, ''B'', ''C''₄, ''C'', ''BC''₄, (''BC''₄, ''C''₄), (''C'', ''BC''₄), (''B''₄, ''B''), (''C'', ''C''₄), ''D''₄, ''F''₄, ''F''.
:Rank 5: ''A''₅, ''B''₅, ''B'', ''C''₅, ''C'', ''BC''₅, (''BC''₅, ''C''₅), (''C'', ''BC''₅), (''B''₅, ''B''), (''C'', ''C''₅), ''D''₅.
:Rank 6: ''A''₆, ''B''₆, ''B'', ''C''₆, ''C'', ''BC''₆, (''BC''₆, ''C''₆), (''C'', ''BC''₆), (''B''₆, ''B''), (''C'', ''C''₆), ''D''₆, ''E''₆,
:Rank 7: ''A''₇, ''B''₇, ''B'', ''C''₇, ''C'', ''BC''₇, (''BC''₇, ''C''₇), (''C'', ''BC''₇), (''B''₇, ''B''), (''C'', ''C''₇), ''D''₇, ''E''₇,
:Rank 8: ''A''₈, ''B''₈, ''B'', ''C''₈, ''C'', ''BC''₈, (''BC''₈, ''C''₈), (''C'', ''BC''₈), (''B''₈, ''B''), (''C'', ''C''₈), ''D''₈, ''E''₈,
:Rank ''n'' (''n''>8): ''A''_''n'', ''B''_''n'', ''B'', ''C''_''n'', ''C'', ''BC''_''n'', (''BC''_''n'', ''C''_''n''), (''C'', ''BC''_''n''), (''B''_''n'', ''B''), (''C'', ''C''_''n''), ''D''_''n''.

Applications

* showed that the affine root systems index Macdonald identities
* 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.
* showed that affine roots systems index families of Macdonald polynomials.

References

*
*
*{{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

Discrete groups
Lie algebras
Orthogonal polynomials