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B2 and C2, identical by a 45 degree rotation, each with 4 short roots, and 4 long ones. | |
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A2, with 6 roots. | G2, with 6 short roots and 6 long roots. |
In mathematics, a root space diagram is a geometric diagram showing the root system vectors in a Euclidean space satisfying certain geometrical properties.
The root system of the simply-laced Lie groups, , , 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 family root systems correspond to the vertices of an expanded n- simplex. The family root system corresponds to the vertices of a rectified n- orthoplex. The root systems correspond to the 122, 231, and 421 uniform polytopes respectively.
For the nonsimply-laced groups, , , and 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 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 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 and 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 group have the 2n vertices of the n-orthoplex as short vectors.
The nonsimply-laced groups can also be seen as Geometric folding of higher rank simply-laced groups. is a folding of , and is a folding of . is a folding of and is a folding of . The folding as seen as an orthogonal projection changes equal length vectors outside the projective subspace to become shortened, expressing the short roots.
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...).
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...).
The 240 roots of E8 can be constructed in two sets: 112 (22×8C2) with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from 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.
The 48 roots of F4 can be constructed in three sets: 24 with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, 8 with coordinates permuted from , and 16 roots with coordinates from from .
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.
Lie group | ||||||
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Diagrams |
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Diagrams II |
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Polygon | square | Hexagon | Square+square | Hexagon+hexagon | ||
Coxeter diagram | ![]() ![]() ![]() |
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Roots | 4 | 6 | 4+4 | 6+6 | ||
Dimensions | 6 | 8 | 10 | 14 | ||
Symmetry order | 4 | 6 | 8 | 12 | ||
Dynkin diagram | ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
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.
Lie group | = | |||||
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Diagrams |
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Diagrams II |
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Polyhedron | Octahedron | Hexagonal bipyramid | Cuboctahedron | cuboctahedron and octahedron | ||
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
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Roots | 6 | 8 | 12 | 6+12 | 12+6 | |
Dimensions | 9 | 11 | 15 | 21 | ||
Symmetry order | 8 | 12 | 24 | 48 | ||
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
8 3A1 roots | 12 A3 roots | ||||||||||||||||||||
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Coxeter plane |
BC3 plane | BC2 plane |
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B4 roots |
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C4 roots |
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[6] | [4] |
There are four unnconnected orthogonal subgroups:
Lie group | 4A1 | A4 = E4 | D4 | B4 | C4 | F4 |
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Projective diagram |
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Polytope | 16-cell | Runcinated 5-cell | Rectified 16-cell | Rectified 16-cell and 16-cell | 24-cell and dual | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 8 | 20 | 24 | 8+24 | 24+8 | 24+24 |
Dimensions | 12 | 24 | 28 | 36 | 36 | 52 |
Symmetry order | 16 | 24 | 192 | 384 | 1152 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
BC4 plane | BC3/D4/A2 plane | BC2/D3 plane | A3 plane |
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[8] | [6] | [4] | [4] |
Coxeter plane |
A4 plane | A3 plane | A2 plane |
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Diagram |
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Plane symmetry |
[[5]]=[10] | [4] | [[3]]=[6] |
Coxeter plane |
BC4 plane | BC3 plane | BC2 plane | A3 plane | F4 plane |
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B4 |
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C4 |
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Plane symmetry |
[8] | [6] | [4] | [4] | [12/3] |
F4 plane | BC4 plane | D4/BC3 plane | A2 plane | D3/BC2/A3 plane | |
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[12] | [8] | [6] | [6] | [4] | |
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Coxeter plane |
F4 plane | BC4 plane | BC3/A2 plane | BC2/A3 plane |
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Diagram |
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Plane symmetry |
[12] | [8] | [6] | [4] |
Others with orthogonal subgroups are generated by a sum of roots from each subgroup, including:
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Lie group | 5A1 | A5 | D5 = E5 | B5 | C5 |
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Projective diagram |
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Polytope | 5-orthoplex | Expanded 5-simplex | Rectified 5-orthoplex | Rectified 5-orthoplex and 5-orthoplex | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 10 | 30 | 40 | 10+40 | 40+10 |
Dimensions | 15 | 35 | 45 | 55 | 55 |
Symmetry order | 32 | 120 | 1920 | 3840 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||||
Simple roots |
A5 plane | A4 plane | A3 plane | A2 plane |
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[6] | [[5]]=[10] | [4] | [[3]]=[6] |
Coxeter plane |
BC5 plane | BC4 plane | BC3 plane | BC2 plane | A3 plane | F4 plane |
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B5 roots |
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C5 roots |
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Plane symmetry |
[10] | [8] | [6] | [4] | [4] | [12/3] |
BC5/A4 plane | BC4/D5 plane | BC3/A2 plane | BC2 plane | A3 plane |
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[10] | [8] | [6] | [4] | [4] |
BC5 plane | BC4/D5 plane | BC3/D4/A2 plane | BC2/D3 plane | A3 plane |
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[10] | [8] | [6] | [4] | [4] |
Six dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:
Lie group | 6A1 | A6 | D6 |
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Projective diagram |
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Polytope | 6-orthoplex | Expanded 6-simplex | Rectified 6-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 12 | 42 | 60 |
Dimensions | 18 | 48 | 66 |
Symmetry order | 64 | 720 | 23040 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B6 | C6 | E6 |
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Projective diagram |
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Polytope | Rectified 6-orthoplex and 6-orthoplex | 122 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 12+60 | 60+12 | 72 |
Dimensions | 78 | 78 | 78 |
Symmetry order | 46080 | 51840 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
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[[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC6 plane | BC5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A3 plane |
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[12] | [10] | [8] | [6] | [4] | [6] | [4] |
E6/F4 plane | B5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | A5 plane | BC6 plane | BC2/D3/A3 plane |
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[12] | [10] | [8] | [6] | [6] | [12/2] | [4] |
Seven dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:
Lie group | 7A1 | A7 | D7 |
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Projective diagram |
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Polytope | 7-orthoplex | Expanded 7-simplex | Rectified 7-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 14 | 56 | 84 |
Dimensions | 21 | 63 | 91 |
Symmetry order | 128 | 5040 | 322560 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B7 | C7 | E7 |
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Projective diagram |
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Polytope | Rectified 7-orthoplex and 7-orthoplex | 231 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 14+84 | 84+14 | 126 |
Dimensions | 105 | 105 | 133 |
Symmetry order | 645,120 | 2,903,040 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots | : |
A7 plane | A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
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[8] | [[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC7 plane | BC6/D7 plane | BC5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A3 plane |
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[14] | [12] | [10] | [8] | [6] | [4] | [6] | [4] |
E7 | E6/F4 plane | A6/BC7 plane | A5 plane | D7/BC6 plane |
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[18] | [12] | [7x2] | [6] | [12/2] |
A4/BC5/D6 plane | D5/BC4 plane | A2/BC3/D4 plane | A3/BC2/D3 plane | |
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[10] | [8] | [6] | [4] |
Eight dimensional root systems in Coxeter plane orthographic projections:
Lie group | 8A1 | A8 | D8 |
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Projective diagram |
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Polytope | 8-orthoplex | Expanded 8-simplex | Rectified 8-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 16 | 72 | 112 |
Dimensions | 24 | 80 | 120 |
Symmetry order | 256 | 40,320 | 5,160,960 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B8 | C8 | E8 |
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Projective diagram |
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Polytope | Rectified 8-orthoplex and 8-orthoplex | 421 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 16+112 | 112+16 | 112+128 |
Dimensions | 136 | 136 | 248 |
Symmetry order | 10,321,920 | 696,729,600 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
A8 plane | A7 plane | A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
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[[9]]=[18] | [8] | [[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC8 plane | BC7/D8 plane | BC6/D7 plane | BC5/D6/A4 plane | BC4/D5 plane |
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[16] | [14] | [12] | [10] | [8] |
BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A7 plane | A3 plane |
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[6] | [4] | [8] | [6] | [4] |
The split real forms for the complex semisimple Lie algebras are: [1]
These are the Lie algebras of the split real groups of the complex Lie groups.
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.
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B2 and C2, identical by a 45 degree rotation, each with 4 short roots, and 4 long ones. | |
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A2, with 6 roots. | G2, with 6 short roots and 6 long roots. |
In mathematics, a root space diagram is a geometric diagram showing the root system vectors in a Euclidean space satisfying certain geometrical properties.
The root system of the simply-laced Lie groups, , , 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 family root systems correspond to the vertices of an expanded n- simplex. The family root system corresponds to the vertices of a rectified n- orthoplex. The root systems correspond to the 122, 231, and 421 uniform polytopes respectively.
For the nonsimply-laced groups, , , and 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 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 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 and 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 group have the 2n vertices of the n-orthoplex as short vectors.
The nonsimply-laced groups can also be seen as Geometric folding of higher rank simply-laced groups. is a folding of , and is a folding of . is a folding of and is a folding of . The folding as seen as an orthogonal projection changes equal length vectors outside the projective subspace to become shortened, expressing the short roots.
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...).
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...).
The 240 roots of E8 can be constructed in two sets: 112 (22×8C2) with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from 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.
The 48 roots of F4 can be constructed in three sets: 24 with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, 8 with coordinates permuted from , and 16 roots with coordinates from from .
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.
Lie group | ||||||
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Diagrams |
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Diagrams II |
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Polygon | square | Hexagon | Square+square | Hexagon+hexagon | ||
Coxeter diagram | ![]() ![]() ![]() |
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Roots | 4 | 6 | 4+4 | 6+6 | ||
Dimensions | 6 | 8 | 10 | 14 | ||
Symmetry order | 4 | 6 | 8 | 12 | ||
Dynkin diagram | ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
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.
Lie group | = | |||||
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Diagrams |
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Diagrams II |
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Polyhedron | Octahedron | Hexagonal bipyramid | Cuboctahedron | cuboctahedron and octahedron | ||
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
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Roots | 6 | 8 | 12 | 6+12 | 12+6 | |
Dimensions | 9 | 11 | 15 | 21 | ||
Symmetry order | 8 | 12 | 24 | 48 | ||
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
8 3A1 roots | 12 A3 roots | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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Coxeter plane |
BC3 plane | BC2 plane |
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B4 roots |
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C4 roots |
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[6] | [4] |
There are four unnconnected orthogonal subgroups:
Lie group | 4A1 | A4 = E4 | D4 | B4 | C4 | F4 |
---|---|---|---|---|---|---|
Projective diagram |
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Polytope | 16-cell | Runcinated 5-cell | Rectified 16-cell | Rectified 16-cell and 16-cell | 24-cell and dual | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 8 | 20 | 24 | 8+24 | 24+8 | 24+24 |
Dimensions | 12 | 24 | 28 | 36 | 36 | 52 |
Symmetry order | 16 | 24 | 192 | 384 | 1152 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | ||||||
Simple roots |
BC4 plane | BC3/D4/A2 plane | BC2/D3 plane | A3 plane |
---|---|---|---|
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[8] | [6] | [4] | [4] |
Coxeter plane |
A4 plane | A3 plane | A2 plane |
---|---|---|---|
Diagram |
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Plane symmetry |
[[5]]=[10] | [4] | [[3]]=[6] |
Coxeter plane |
BC4 plane | BC3 plane | BC2 plane | A3 plane | F4 plane |
---|---|---|---|---|---|
B4 |
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C4 |
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Plane symmetry |
[8] | [6] | [4] | [4] | [12/3] |
F4 plane | BC4 plane | D4/BC3 plane | A2 plane | D3/BC2/A3 plane | |
---|---|---|---|---|---|
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[12] | [8] | [6] | [6] | [4] | |
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Coxeter plane |
F4 plane | BC4 plane | BC3/A2 plane | BC2/A3 plane |
---|---|---|---|---|
Diagram |
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Plane symmetry |
[12] | [8] | [6] | [4] |
Others with orthogonal subgroups are generated by a sum of roots from each subgroup, including:
|
|
Lie group | 5A1 | A5 | D5 = E5 | B5 | C5 |
---|---|---|---|---|---|
Projective diagram |
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Polytope | 5-orthoplex | Expanded 5-simplex | Rectified 5-orthoplex | Rectified 5-orthoplex and 5-orthoplex | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 10 | 30 | 40 | 10+40 | 40+10 |
Dimensions | 15 | 35 | 45 | 55 | 55 |
Symmetry order | 32 | 120 | 1920 | 3840 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||||
Simple roots |
A5 plane | A4 plane | A3 plane | A2 plane |
---|---|---|---|
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[6] | [[5]]=[10] | [4] | [[3]]=[6] |
Coxeter plane |
BC5 plane | BC4 plane | BC3 plane | BC2 plane | A3 plane | F4 plane |
---|---|---|---|---|---|---|
B5 roots |
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C5 roots |
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Plane symmetry |
[10] | [8] | [6] | [4] | [4] | [12/3] |
BC5/A4 plane | BC4/D5 plane | BC3/A2 plane | BC2 plane | A3 plane |
---|---|---|---|---|
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[10] | [8] | [6] | [4] | [4] |
BC5 plane | BC4/D5 plane | BC3/D4/A2 plane | BC2/D3 plane | A3 plane |
---|---|---|---|---|
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[10] | [8] | [6] | [4] | [4] |
Six dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:
Lie group | 6A1 | A6 | D6 |
---|---|---|---|
Projective diagram |
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Polytope | 6-orthoplex | Expanded 6-simplex | Rectified 6-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 12 | 42 | 60 |
Dimensions | 18 | 48 | 66 |
Symmetry order | 64 | 720 | 23040 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B6 | C6 | E6 |
---|---|---|---|
Projective diagram |
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Polytope | Rectified 6-orthoplex and 6-orthoplex | 122 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 12+60 | 60+12 | 72 |
Dimensions | 78 | 78 | 78 |
Symmetry order | 46080 | 51840 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
---|---|---|---|---|
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[[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC6 plane | BC5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A3 plane |
---|---|---|---|---|---|---|
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[12] | [10] | [8] | [6] | [4] | [6] | [4] |
E6/F4 plane | B5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | A5 plane | BC6 plane | BC2/D3/A3 plane |
---|---|---|---|---|---|---|
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[12] | [10] | [8] | [6] | [6] | [12/2] | [4] |
Seven dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:
Lie group | 7A1 | A7 | D7 |
---|---|---|---|
Projective diagram |
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Polytope | 7-orthoplex | Expanded 7-simplex | Rectified 7-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 14 | 56 | 84 |
Dimensions | 21 | 63 | 91 |
Symmetry order | 128 | 5040 | 322560 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B7 | C7 | E7 |
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Projective diagram |
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Polytope | Rectified 7-orthoplex and 7-orthoplex | 231 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 14+84 | 84+14 | 126 |
Dimensions | 105 | 105 | 133 |
Symmetry order | 645,120 | 2,903,040 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots | : |
A7 plane | A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
---|---|---|---|---|---|
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[8] | [[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC7 plane | BC6/D7 plane | BC5/D6/A4 plane | BC4/D5 plane | BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A3 plane |
---|---|---|---|---|---|---|---|
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[14] | [12] | [10] | [8] | [6] | [4] | [6] | [4] |
E7 | E6/F4 plane | A6/BC7 plane | A5 plane | D7/BC6 plane |
---|---|---|---|---|
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[18] | [12] | [7x2] | [6] | [12/2] |
A4/BC5/D6 plane | D5/BC4 plane | A2/BC3/D4 plane | A3/BC2/D3 plane | |
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[10] | [8] | [6] | [4] |
Eight dimensional root systems in Coxeter plane orthographic projections:
Lie group | 8A1 | A8 | D8 |
---|---|---|---|
Projective diagram |
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Polytope | 8-orthoplex | Expanded 8-simplex | Rectified 8-orthoplex |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 16 | 72 | 112 |
Dimensions | 24 | 80 | 120 |
Symmetry order | 256 | 40,320 | 5,160,960 |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
Lie group | B8 | C8 | E8 |
---|---|---|---|
Projective diagram |
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Polytope | Rectified 8-orthoplex and 8-orthoplex | 421 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Roots | 16+112 | 112+16 | 112+128 |
Dimensions | 136 | 136 | 248 |
Symmetry order | 10,321,920 | 696,729,600 | |
Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Cartan matrix | |||
Simple roots |
A8 plane | A7 plane | A6 plane | A5 plane | A4 plane | A3 plane | A2 plane |
---|---|---|---|---|---|---|
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[[9]]=[18] | [8] | [[7]]=[14] | [6] | [[5]]=[10] | [4] | [[3]]=[6] |
BC8 plane | BC7/D8 plane | BC6/D7 plane | BC5/D6/A4 plane | BC4/D5 plane |
---|---|---|---|---|
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[16] | [14] | [12] | [10] | [8] |
BC3/D4/G2/A2 plane | BC2/D3 plane | A5 plane | A7 plane | A3 plane |
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[6] | [4] | [8] | [6] | [4] |
The split real forms for the complex semisimple Lie algebras are: [1]
These are the Lie algebras of the split real groups of the complex Lie groups.
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.