User:Tomruen/Root space diagram

In mathematics, a root space diagram is a geometric diagram showing the root system vectors in a Euclidean space satisfying certain geometrical properties.

Construction
The root system of the simply-laced Lie groups, $$A_n$$, $$D_n$$, $$E_n$$ correspond to vertices of specific uniform polytopes of the same symmetry group. A root space diagram corresponds to projected images of these polytope vertices. The $$A_n$$ family root systems correspond to the vertices of an expanded n-simplex. The $$D_n$$ family root system corresponds to the vertices of a rectified n-orthoplex. The $$E_6, E_7, E_8$$ root systems correspond to the 122, 231, and 421 uniform polytopes respectively.

For the nonsimply-laced groups, $$B_n$$, $$C_n$$, $$G_2$$ and $$F_4$$ contain the vertices of two uniform polytopes of different sizes and the same center, each polytype vertices corresponding to either the short or long root vectors. The $$G_2$$ group can be seen as the vertices of two sets of 6 vertices from two regular hexagons, with the vertices of the second hexagon at the mid-edges of the first hexagon. The $$F_4$$ group root can be seen as 2 sets of 24 vertices from the 24-cell in dual positions, with the vertices of the second 24-cell being at the tetrahedral facet centers of the first. Finally the $$B_n$$ and $$C_n$$ root systems can be seen as the vertices of an n-orthoplex, and a rectified n-orthoplex, alternating which set of vertices are the short and long ones. The $$B_n$$ group have the 2n vertices of the n-orthoplex as short vectors.

Construction from folding
The nonsimply-laced groups can also be seen as Geometric folding of higher rank simply-laced groups. $$G_2$$ is a folding of $$D_4$$, and $$F_4$$ is a folding of $$E_6$$. $$C_n$$ is a folding of $$A_{2n-1}$$ and $$B_n$$ is a folding of $$D_{n+1}$$. The folding as seen as an orthogonal projection changes equal length vectors outside the projective subspace to become shortened, expressing the short roots.

A family
The An root system can be seen as vertices of an expanded n-simplex. These roots can be seen as positioned by all permutations of coordinates of (1,-1,0,0,0...) in (n+1) space, with a hyperplane normal vector of (1,1,1...).

D family
The Dn root system can be seen in the vertices of a rectified n-orthoplex, coordinates all sign and coordinate permutations of (1,1,0,0...). These vertices exist in 3 hyperplanes, with a rectified n-simplex as facets on two opposite sides (-1,-1,0,0...) and (1,1,0,0,0...), and a middle hyperplane with the vertex arrangement of a expanded n-simplex as coordinate permutations of (1,-1,0,0,0...).

E family
The 240 roots of E8 can be constructed in two sets: 112 (22×8C2) with coordinates obtained from $$(\pm 2,\pm 2,0,0,0,0,0,0)\,$$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from $$(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,$$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

The E7 and E6 roots can be seen as subspaces of 8-space above.

F4
The 48 roots of F4 can be constructed in three sets: 24 with coordinates obtained from $$(\pm 2,\pm 2,0,0)\,$$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, 8 with coordinates permuted from $$(\pm 2,0,0,0)\,$$, and 16 roots with coordinates from from $$(\pm 1,\pm 1,\pm 1,\pm 1)\,$$.

Rank 2 systems
In the second set of diagrams, the roots are drawn as red circle symbols around an origin. The edges drawn correspond to the shortest edges of the corresponding polygons. In higher dimensional graphs roots may be overlapping in space in an orthogonal projection, so different colors are used by the order of overlap.

Rank 3 systems
Rank 3 systems exist in 3-space, and can be drawn as oblique projection. Root system B3, C3, and A3=D3 as points within a cuboctahedron and octahedron.

Nonsimple groups
There are four unnconnected orthogonal subgroups:
 * 1)  - $$3 A_1$$ - 6 roots (2×3)
 * 2)  - $$A_2 + A_1$$ - 8 roots (6+2)
 * 3)  - $$B_2 + A_1$$ - 10 roots (8+2)
 * 4)  - $$G_2 + A_1$$ - 14 roots (12+2)

Nonsimple groups
Others with orthogonal subgroups are generated by a sum of roots from each subgroup, including:

Rank 6 systems
Six dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:

Rank 7 systems
Seven dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:

Rank 8 systems
Eight dimensional root systems in Coxeter plane orthographic projections:

Classical Lie groups
Related classical Lie groups:
 * SU(n+1)=An
 * SO(2n+1)=Bn
 * SO(2n)=Dn
 * Sp(2n)=Cn

The split real forms for the complex semisimple Lie algebras are: These are the Lie algebras of the split real groups of the complex Lie groups.
 * $$A_n, \mathfrak{sl}_{n+1}(\mathbf{C}): \mathfrak{sl}_{n+1}(\mathbf{R})$$
 * $$B_n, \mathfrak{so}_{2n+1}(\mathbf{C}): \mathfrak{so}_{n,n+1}(\mathbf{R})$$
 * $$C_n, \mathfrak{sp}_n(\mathbf{C}): \mathfrak{sp}_n(\mathbf{R})$$
 * $$D_n, \mathfrak{so}_{2n}(\mathbf{C}): \mathfrak{so}_{n,n}(\mathbf{R})$$
 * Exceptional Lie algebras: $$E_6, E_7, E_8, F_4, G_2$$ have split real forms EI, EV, EVIII, FI, G.

Note that for sl and sp, the real form is the real points of (the Lie algebra of) the same algebraic group, while for so one must use the split forms (of maximally indefinite index), as SO is compact.

Related lattices/honeycombs

 * $${\tilde{A}}_n$$, An lattice: Simplectic honeycomb, {3[n]}
 * $${\tilde{D}}_n$$, Dn lattice: Demicubic honeycomb, {31,1,3n-4,4}
 * $${\tilde{E}}_6$$, E6 lattice: 222 honeycomb, {32,2,2}
 * $${\tilde{E}}_7$$, E7 lattice: 331 honeycomb, {33,3,1}
 * $${\tilde{E}}_8$$, E8 lattice: 521 honeycomb, {35,2,1}