Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.
Finally, root systems are important for their own sake, as in spectral graph theory.
[1] As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots.
The rank of a root system Φ is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space.
The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem[10]).
[11] He used them in his attempt to classify all simple Lie algebras over the field of complex numbers.
(Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4.
Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.
The set of integral elements is called the weight lattice associated to the given root system.
The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).
The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber.
The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines.
A related result is this one:[23] Irreducible root systems classify a number of related objects in Lie theory, notably the following: In each case, the roots are non-zero weights of the adjoint representation.
We now give a brief indication of how irreducible root systems classify simple Lie algebras over
[24] A preliminary result says that a semisimple Lie algebra is simple if and only if the associated root system is irreducible.
[25] We thus restrict attention to irreducible root systems and simple Lie algebras.
Irreducible root systems are named according to their corresponding connected Dynkin diagrams.
[30] In general, the An root lattice is the vertex arrangement of the n-dimensional simplicial honeycomb.
The reflection σn through the hyperplane perpendicular to the short root αn is of course simply negation of the nth coordinate.
C2 is isomorphic to B2 via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.
Reflection through the hyperplane perpendicular to αn is the same as transposing and negating the adjacent n-th and (n − 1)-th coordinates.
The twelve D3 root vectors are expressed as the vertices of , a lower symmetry construction of the cuboctahedron.
The twenty-four D4 root vectors are expressed as the vertices of , a lower symmetry construction of the 24-cell.
One choice of simple roots for E8 in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is: (the above choice of simple roots for D7) along with
One choice of simple roots for E8 in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is (the above choice of simple roots for A7) along with (Using β3 would give an isomorphic result.
This facilitates explicit definitions of E7 and E6 as Note that deleting α1 and then α2 gives sets of simple roots for E7 and E6.
However, these sets of simple roots are in different E7 and E6 subspaces of E8 than the ones written above, since they are not orthogonal to α1 or α2.
For F4, let E = R4, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd.
, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset.
Modulo reflection, for a given α there are only 5 nontrivial possibilities for β , and 3 possible angles between α and β in a set of simple roots. Subscript letters correspond to the series of root systems for which the given β can serve as the first root and α as the second root (or in F 4 as the middle 2 roots).
