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 (Macdonald 1972) and (Bruhat Tits) (except that both these papers accidentally omitted the Dynkin diagram Script error: No such module "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]\displaystyle{ u,v \in E }[/math], then it is well defined an element in V denoted as [math]\displaystyle{ u-v }[/math] which is the only element w such that [math]\displaystyle{ v+w=u }[/math].
Now suppose we have a scalar product [math]\displaystyle{ (\cdot,\cdot) }[/math] on V. This defines a metric on E as [math]\displaystyle{ d(u,v)=\vert(u-v,u-v)\vert }[/math].
Consider the vector space F of affine-linear functions [math]\displaystyle{ f\colon E\longrightarrow \mathbb{R} }[/math]. Having fixed a [math]\displaystyle{ x_0\in E }[/math], every element in F can be written as [math]\displaystyle{ f(x)=Df(x-x_0)+f(x_0) }[/math] with [math]\displaystyle{ Df }[/math] a linear function on V that doesn't depend on the choice of [math]\displaystyle{ 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]\displaystyle{ (f,g)=(Df,Dg) }[/math]. Set [math]\displaystyle{ f^\vee =\frac{2f}{(f,f)} }[/math] and [math]\displaystyle{ v^\vee =\frac{2v}{(v,v)} }[/math] for any [math]\displaystyle{ f\in F }[/math] and [math]\displaystyle{ v\in V }[/math] respectively. The identification let us define a reflection [math]\displaystyle{ w_f }[/math] over E in the following way:
- [math]\displaystyle{ w_f(x)=x-f^\vee(x)Df }[/math]
By transposition [math]\displaystyle{ w_f }[/math] acts also on F as
- [math]\displaystyle{ w_f(g)=g-(f^\vee,g)f }[/math]
An affine root system is a subset [math]\displaystyle{ S\in F }[/math] such that:
- S spans F and its elements are non-constant.
- [math]\displaystyle{ w_a(S)=S }[/math] for every [math]\displaystyle{ a\in S }[/math].
- [math]\displaystyle{ (a,b^\vee)\in\mathbb{Z} }[/math] for every [math]\displaystyle{ a,b\in S }[/math].
The elements of S are called affine roots. Denote with [math]\displaystyle{ w(S) }[/math] the group generated by the [math]\displaystyle{ w_a }[/math] with [math]\displaystyle{ a\in S }[/math]. We also ask
- [math]\displaystyle{ w(S) }[/math] as a discrete group acts properly on E.
This means that for any two compacts [math]\displaystyle{ K,H\subseteq E }[/math] the elements of [math]\displaystyle{ w(S) }[/math] such that [math]\displaystyle{ w(K)\cap H\neq \varnothing }[/math] 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.
Irreducible affine root systems by rank
- Rank 1: A1, BC1, (BC1, C1), (C∨1, BC1), (C∨1, C1).
- Rank 2: A2, C2, C∨2, BC2, (BC2, C2), (C∨2, BC2), (B2, B∨2), (C∨2, C2), G2, G∨2.
- Rank 3: A3, B3, B∨3, C3, C∨3, BC3, (BC3, C3), (C∨3, BC3), (B3, B∨3), (C∨3, C3).
- Rank 4: A4, B4, B∨4, C4, C∨4, BC4, (BC4, C4), (C∨4, BC4), (B4, B∨4), (C∨4, C4), D4, F4, F∨4.
- Rank 5: A5, B5, B∨5, C5, C∨5, BC5, (BC5, C5), (C∨5, BC5), (B5, B∨5), (C∨5, C5), D5.
- Rank 6: A6, B6, B∨6, C6, C∨6, BC6, (BC6, C6), (C∨6, BC6), (B6, B∨6), (C∨6, C6), D6, E6,
- Rank 7: A7, B7, B∨7, C7, C∨7, BC7, (BC7, C7), (C∨7, BC7), (B7, B∨7), (C∨7, C7), D7, E7,
- Rank 8: A8, B8, B∨8, C8, C∨8, BC8, (BC8, C8), (C∨8, BC8), (B8, B∨8), (C∨8, C8), D8, E8,
- Rank n (n>8): An, Bn, B∨n, Cn, C∨n, BCn, (BCn, Cn), (C∨n, BCn), (Bn, B∨n), (C∨n, Cn), Dn.
Applications
- (Macdonald 1972) showed that the affine root systems index Macdonald identities
- (Bruhat Tits) 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, http://www.numdam.org/item?id=PMIHES_1972__41__5_0
- Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae 15 (2): 91–143, doi:10.1007/BF01418931, ISSN 0020-9910, Bibcode: 1971InMat..15...91M
- Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9
Original source: https://en.wikipedia.org/wiki/Affine root system.
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