Английская Википедия: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 Шаблон:Harvtxt and Шаблон:Harvtxt (except that both these papers accidentally omitted the Dynkin diagram Шаблон: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 <math> u,v \in E</math>, then it is well defined an element in V denoted as <math>u-v</math> which is the only element w such that <math>v+w=u</math>.
Now suppose we have a scalar product <math>(\cdot,\cdot)</math> on V. This defines a metric on E as <math> d(u,v)=\vert(u-v,u-v)\vert</math>.
Consider the vector space F of affine-linear functions <math>f\colon E\longrightarrow \mathbb{R}</math>. Having fixed a <math>x_0\in E</math>, every element in F can be written as <math>f(x)=Df(x-x_0)+f(x_0)</math> with <math>Df</math> a linear function on V that doesn't depend on the choice of <math>x_0</math>.
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 <math>(f,g)=(Df,Dg)</math>. Set <math>f^\vee =\frac{2f}{(f,f)}</math> and <math>v^\vee =\frac{2v}{(v,v)}</math> for any <math>f\in F</math> and <math>v\in V</math> respectively. The identification let us define a reflection <math>w_f</math> over E in the following way:
- <math> w_f(x)=x-f^\vee(x)Df</math>
By transposition <math>w_f</math> acts also on F as
- <math>w_f(g)=g-(f^\vee,g)f</math>
An affine root system is a subset <math>S\in F</math> such that: Шаблон:Ordered list The elements of S are called affine roots. Denote with <math>w(S)</math> the group generated by the <math>w_a</math> with <math>a\in S</math>. We also ask Шаблон:Ordered list This means that for any two compacts <math>K,H\subseteq E</math> the elements of <math>w(S)</math> such that <math>w(K)\cap H\neq \varnothing</math> are a finite number.
Classification
The affine roots systems A1 = B1 = BШаблон:Su = C1 = CШаблон:Su are the same, as are the pairs B2 = C2, BШаблон:Su = CШаблон: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.
Irreducible affine root systems by rank
- Rank 1: A1, BC1, (BC1, C1), (CШаблон:Su, BC1), (CШаблон:Su, C1).
- Rank 2: A2, C2, CШаблон:Su, BC2, (BC2, C2), (CШаблон:Su, BC2), (B2, BШаблон:Su), (CШаблон:Su, C2), G2, GШаблон:Su.
- Rank 3: A3, B3, BШаблон:Su, C3, CШаблон:Su, BC3, (BC3, C3), (CШаблон:Su, BC3), (B3, BШаблон:Su), (CШаблон:Su, C3).
- Rank 4: A4, B4, BШаблон:Su, C4, CШаблон:Su, BC4, (BC4, C4), (CШаблон:Su, BC4), (B4, BШаблон:Su), (CШаблон:Su, C4), D4, F4, FШаблон:Su.
- Rank 5: A5, B5, BШаблон:Su, C5, CШаблон:Su, BC5, (BC5, C5), (CШаблон:Su, BC5), (B5, BШаблон:Su), (CШаблон:Su, C5), D5.
- Rank 6: A6, B6, BШаблон:Su, C6, CШаблон:Su, BC6, (BC6, C6), (CШаблон:Su, BC6), (B6, BШаблон:Su), (CШаблон:Su, C6), D6, E6,
- Rank 7: A7, B7, BШаблон:Su, C7, CШаблон:Su, BC7, (BC7, C7), (CШаблон:Su, BC7), (B7, BШаблон:Su), (CШаблон:Su, C7), D7, E7,
- Rank 8: A8, B8, BШаблон:Su, C8, CШаблон:Su, BC8, (BC8, C8), (CШаблон:Su, BC8), (B8, BШаблон:Su), (CШаблон:Su, C8), D8, E8,
- Rank n (n>8): An, Bn, BШаблон:Su, Cn, CШаблон:Su, BCn, (BCn, Cn), (CШаблон:Su, BCn), (Bn, BШаблон:Su), (CШаблон:Su, Cn), Dn.
Applications
- Шаблон:Harvtxt showed that the affine root systems index Macdonald identities
- Шаблон: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.
- Шаблон:Harvtxt showed that affine roots systems index families of Macdonald polynomials.
References
