Dynkin diagram

The Dynkin diagram or Dynkin diagram named after Evgeny Borisovich Dynkin is a type of graph in which some edges are doubled or tripled (drawn as a double or triple line). Multiple edges, with some limitations, are oriented .

The main interest in Dynkin diagrams is that they allow us to introduce a classification of semisimple Lie algebras over algebraically closed fields . The diagrams lead to Weyl groups , that is, to many (although not all) . Dynkin diagrams also arise in other contexts.

The term “Dynkin diagram” can be ambiguous. In some cases, Dynkin diagrams are assumed to be oriented, in which case they correspond to root systems and semisimple Lie algebras, while in other cases they are assumed to be non-oriented, and then they correspond to Weyl groups. Oriented Charts for $B_{n}$ ${\ displaystyle B_ {n}}$ $B_ {n}$ and $C_{n}$ ${\ displaystyle C_ {n}}$ $C_ {n}$ give the same undirected diagram that is denoted $BC_{n}.$ ${\ displaystyle BC_ {n}.}$ $BC_ {n}.$ In this article, by default, the “Dynkin diagram” means the oriented Dynkin diagram, and for undirected Dynkin diagrams this is indicated explicitly.

The final diagrams of Dynkin
Affinity (extended) diagrams of Dynkin

Content

Classification of Semisimple Lie Algebras

The fundamental interest in Dynkin diagrams arises because they allow us to classify semisimple Lie algebras over algebraically closed fields. Some classify such Lie algebras through their root systems , which can be represented by Dynkin diagrams. Others classify Dynkin diagrams according to the constraints that they must satisfy, as described below.

Getting rid of the direction of the edges of the graph corresponds to replacing the root system with a , which they create, the so-called Weyl group , and thus the unoriented Dynkin diagrams classify the Weyl groups.

Related Classifications

Dynkin diagrams can be used to classify many different objects, and the notation “A _n , B _n , ...” is used to refer to all such interpretations depending on the context. Such ambiguity can be confusing.

The central classification refers to simple Lie algebras that have a root system and with which (oriented) Dynkin diagrams are associated. All three (of the following), for example, can be designated as B _n .

The non- oriented Dynkin diagram is a kind of Coxeter diagram and corresponds to the Weyl group, which is the associated with the root system. Thus, B _n can refer to an undirected diagram (a special kind of Coxeter diagram), a Weyl group (a specific reflection group), or an abstract Weyl group.

Note that while the Weyl group is abstractly isomorphic to the Coxeter group, the concrete isomorphism depends on the order of simple roots. Note that the notation of Dynkin diagrams is standardized, while the Coxeter diagrams and group notation vary and are sometimes consistent with the Dynkin diagram, and sometimes not.

Finally, sometimes associated objects are denoted by the same notation, although this cannot always be done on a regular basis. Examples:

formed by the root system, as in the E ₈ lattice . It is determined naturally, but not one-to-one - for example, A ₂ and G ₂ form the same hexagonal lattice .
An associated polytope - for example, can be designated as “polytope E ₈ ”, since its vertices are obtained from the root system E ₈ and it has a Coxeter group E ₈ as a symmetry group.
Associated quadratic form or manifold - for example, E ₈ has the intersection form defined by the lattice E ₈ .

These latter notations are most often used for objects associated with exceptional diagrams - for objects associated with ordinary diagrams (A, B, C, D), traditional names are used.

Index ( n ) is equal to the number of nodes in the diagram, the number of simple roots in the basis, the dimensions of the root lattice and the linear shell of the root system, the number of generators of the Coxeter group, and the rank of Lie algebra. However, n is not necessarily equal to the dimension of the defining module ( fundamental representation ) of the Lie algebra - the index of the Dynkin diagram should not be confused with the index of the Lie algebra. For example, $B_{4}$ ${\ displaystyle B_ {4}}$ corresponds to ${\mathfrak {so}}_{2\cdot 4+1}={\mathfrak {so}}_{9},$ ${\ displaystyle {\ mathfrak {so}} _ {2 \ cdot 4 + 1} = {\ mathfrak {so}} _ {9},}$ which acts in a 9-dimensional space but has rank 4 as a Lie algebra.

Dynkin diagrams in one thread , that is, those that do not have multiple edges (A, D, E) classify many other mathematical objects. See the discussion in ADE Classification .

Example: A2

A_{2}

{\ displaystyle A_ {2}}

, root system.

For example, the designation $A_{2}$ ${\ displaystyle A_ {2}}$ may relate to:

Dynkin diagram with two connected nodes, , which can also be considered as a Coxeter diagram .
Root system with two simple roots at an angle $2\pi /3$ ${\ displaystyle 2 \ pi / 3}$ (120 degrees).
Algebra Lee ${\mathfrak {sl}}_{2+1}={\mathfrak {sl}}_{3}$ ${\ displaystyle {\ mathfrak {sl}} _ {2 + 1} = {\ mathfrak {sl}} _ {3}}$ 2.
The Weyl group of root symmetries (reflections with respect to hyperplanes orthogonal to the roots), which is isomorphic to the symmetric group $S_{3}$ ${\ displaystyle S_ {3}}$ (about 6).
The Coxeter abstract group , represented by generators and links, $\left\langle r_{1},r_{2}\mid (r_{1})^{2}=(r_{2})^{2}=(r_{i}r_{j})^{3}=1\right\rangle .$ ${\ displaystyle \ left \ langle r_ {1}, r_ {2} \ mid (r_ {1}) ^ {2} = (r_ {2}) ^ {2} = (r_ {i} r_ {j}) ^ {3} = 1 \ right \ rangle.}$

Limitations

Dynkin diagrams must satisfy certain constraints, those that satisfy the finite Coxeter - Dynkin diagrams , and, in addition, additional crystallographic constraints.

Linking Coxeter Charts

Dynkin diagrams are closely related to Coxeter diagrams of Coxeter end groups, and terminology is often combined ^{[note 1]} .

Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects:

Partial focus: Dynkin diagrams are partially oriented — any multiple edge (in Coxeter terms, labeled “4” and above) has a direction (an arrow directed from one node to another). Thus, the Dynkin diagram carries more information than the corresponding Coxeter diagram (undirected graph).; At the level of root systems, the direction corresponds to an indication of a shorter vector. The edges marked with “3” have no direction, since the corresponding vectors must be of equal length. (Note: Some authors use the opposite convention by pointing the arrow to a longer vector.)
Crystallographic limitation: Dynkin diagrams must satisfy an additional restriction, namely, edges with labels 2, 3, 4, and 6 are allowed. This restriction does not apply to Coxeter diagrams, so not every Coxeter diagram of a finite group comes from the Dynkin diagram.; At the root system level, this corresponds .

Another difference, purely stylistic, is that it is customary to draw Dynkin diagrams with double and triple edges between nodes (for p = 4, 6), and not marked with the number “ p ”.

The term “Dynkin diagram” is sometimes referred to as oriented graphs, and sometimes to non-oriented ones . For accuracy, in this article, the “Dynkin diagram” will mean oriented, and the corresponding undirected graph will be called the “undirected Dynkin diagram”. Thus, Dynkin diagrams and Coxeter diagrams can be related as follows:

	crystallographic	point groups
oriented	Dynkin Charts
non-oriented	Undirected Dynkin Charts	Coxeter - Dynkin diagrams of finite groups

This means that Coxeter diagrams of finite groups correspond to point groups generated by reflections, while Dynkin diagrams must satisfy additional constraints corresponding . This also means that Coxeter diagrams are non-oriented, while Dynkin diagrams are (partially) oriented.

Mathematical objects systematized by diagrams:

	crystallographic	point groups
oriented	Root systems
non-oriented	Weil groups	Coxeter end groups

The empty space in the upper right corner corresponding to the oriented graphs with the undirected graphs of any Coxeter diagram (finite group) lying below them can be defined formally, but these definitions do not allow a simple interpretation in terms of mathematical objects.

There are natural narrowing mappings - from Dynkin diagrams to undirected Dynkin diagrams, and, accordingly, from root systems to associated Weyl groups, as well as direct mappings from undirected Dynkin diagrams to Coxeter diagrams, and, accordingly, from Weyl groups to finite Coxeter groups.

Narrowing mappings are mapped to (by definition), but not one-to-one. For example, diagrams B _n and C _{n are} mapped to the same undirected diagram, so sometimes the resulting Coxeter diagram and the Weyl group are denoted by BC _n .

Direct mappings are simply an inclusion - undirected Dynkin diagrams are a special case of Coxeter diagrams, and Weyl groups are special cases of finite Coxeter groups, and this mapping is not on , since not every Coxeter diagram is an undirected Dynkin diagram (missing diagrams are H ₃ , H ₄ and I ₂ ( p ) for p = 5 p ≥ 7), and, accordingly, not every finite Coxeter group is a Weyl group.

Isomorphisms

connected Dynkin diagrams.

Dynkin diagrams are usually numbered so that the list is not redundant - $n\geq 1$ ${\ displaystyle n \ geq 1}$ for $A_{n},$ ${\ displaystyle A_ {n},}$ $n\geq 2$ ${\ displaystyle n \ geq 2}$ for $B_{n},$ ${\ displaystyle B_ {n},}$ $n\geq 3$ ${\ displaystyle n \ geq 3}$ for $C_{n},$ ${\ displaystyle C_ {n},}$ $n\geq 4$ ${\ displaystyle n \ geq 4}$ for $D_{n}$ ${\ displaystyle D_ {n}}$ and $E_{n}$ ${\ displaystyle E_ {n}}$ beginning with $n=6.$ ${\ displaystyle n = 6.}$ Elements of families, however, can also be defined for the lower n, obtaining diagrams and corresponding exceptional isomorphisms of Lie algebras and associated Lie groups.

The easiest way to start is with the cases n = 0 or n = 1, in which all series are isometric and there is a single empty diagram and a diagram with one node. Other isomorphisms of connected Dynkin diagrams:

$A_{1}\cong B_{1}\cong C_{1}$ ${\ displaystyle A_ {1} \ cong B_ {1} \ cong C_ {1}}$
$B_{2}\cong C_{2}$ ${\ displaystyle B_ {2} \ cong C_ {2}}$
$D_{2}\cong A_{1}\times A_{1}$ ${\ displaystyle D_ {2} \ cong A_ {1} \ times A_ {1}}$
$D_{3}\cong A_{3}$ ${\ displaystyle D_ {3} \ cong A_ {3}}$
$E_{3}\cong A_{1}\times A_{2}$ ${\ displaystyle E_ {3} \ cong A_ {1} \ times A_ {2}}$
$E_{4}\cong A_{4}$ ${\ displaystyle E_ {4} \ cong A_ {4}}$
$E_{5}\cong D_{5}$ ${\ displaystyle E_ {5} \ cong D_ {5}}$

These isomorphisms correspond to isomorphisms of simple and semisimple Lie algebras.

Automorphisms

The most symmetric Dynkin diagram is D ₄ , which leads to .

In addition to isomorphisms between different diagrams, some diagrams also have isomorphisms on themselves, that is, “ automorphisms ”. Automorphisms of diagrams correspond to Lie algebra, which means that the group of external automorphisms Out = Aut / Inn is equal to the group of automorphisms of a diagram ^[1] ^[2] ^[3] .

Diagrams having nontrivial automorphisms - A _n ( $n>1$ ${\ displaystyle n> 1}$ ), D _n ( $n>1$ ${\ displaystyle n> 1}$ ) and E ₆ . In all these cases, with the exception of D ₄ , there is one nontrivial automorphism (Out = C ₂ , a cyclic group of order 2), while for D ₄ the automorphism group is a symmetric group of three letters ( S ₃ , order 6) - this phenomenon known as " ". It turns out that all these diagram automorphisms can be represented as symmetries of the traditional diagram diagram in the Euclidean plane, but this is just the result of how they are drawn, and not the structure inherent in diagrams.

A _n

For A _{n, an} automorphism of diagrams is a flipping diagram. The nodes of the diagram are indexed by , which (for A _{n −1} ) are equal $\bigwedge ^{i}C^{n}$ ${\ displaystyle \ bigwedge ^ {i} C ^ {n}}$ for $i=1,\dots ,n$ ${\ displaystyle i = 1, \ dots, n}$ , and the automorphism of the diagram corresponds to duality $\bigwedge ^{i}C^{n}\mapsto \bigwedge ^{n-i}C^{n}.$ ${\ displaystyle \ bigwedge ^ {i} C ^ {n} \ mapsto \ bigwedge ^ {ni} C ^ {n}.}$ Considered as a Lie algebra ${\mathfrak {sl}}_{n+1},$ ${\ displaystyle {\ mathfrak {sl}} _ {n + 1},}$ external automorphism can be expressed as a negative transposition, $T\mapsto -T^{\mathrm {T} }$ ${\ displaystyle T \ mapsto -T ^ {\ mathrm {T}}}$ ^[2] .

D _n .

For D _{n, the} diagram automorphism switches two nodes at the end of Y, and corresponds to the switching of two chiral . Considered as a Lie algebra ${\mathfrak {so}}_{2n},$ ${\ displaystyle {\ mathfrak {so}} _ {2n},}$ an external automorphism can be expressed as conjugation using a matrix from O (2 n ) with determinant −1 ^{[note 2]} . notice, that $\mathrm {A} _{3}\cong \mathrm {D} _{3},$ ${\ displaystyle \ mathrm {A} _ {3} \ cong \ mathrm {D} _ {3},}$ so their automorphisms are the same, while $\mathrm {D} _{2}\cong \mathrm {A} _{1}\times \mathrm {A} _{1}$ ${\ displaystyle \ mathrm {D} _ {2} \ cong \ mathrm {A} _ {1} \ times \ mathrm {A} _ {1}}$ and this diagram is disconnected, so that the automorphism corresponds to switching nodes.

For D _{4, the} fundamental representation is isomorphic to two and the resulting symmetric group of three letters ( S ₃ , or, alternatively, a sixth order dihedral group , Dih ₃ ) corresponds to both automorphisms of the Lie algebra and automorphisms of the diagram.

E ₆ .

The automorphism E ₆ corresponds to the turning of the diagram and can be expressed using Jordan algebras ^[2] .

Disconnected diagrams that correspond to semi simple Lie algebras can have automorphisms obtained by rearranging the components of the diagram.

With characteristic 2, the arrow in F ₄ can be ignored, which creates an additional automorphism of diagrams and the corresponding .

With a positive characteristic, there are additional automorphisms of the diagrams - roughly speaking, with the characteristic p, we can ignore the arrows on the bonds of multiplicity p in the Dynkin diagram when we consider the automorphism of the diagrams. Thus, for characteristic 2, there is an automorphism of order 2 for $\mathrm {B} _{2}\cong \mathrm {C} _{2}$ ${\ displaystyle \ mathrm {B} _ {2} \ cong \ mathrm {C} _ {2}}$ and for F ₄ , while for characteristic 3 there is an automorphism of order 2 for G ₂ .

Construction of Lie groups using diagram automorphisms

Automorphisms of diagrams create additional Lie groups and Lie- type groups , which is the reason for their central importance in the classification of finite simple groups.

The construction of the Chevalley group of Lie groups in terms of their Dynkin diagrams does not give classical groups, namely, unitary groups and . Steinberg groups build ² A _n unitary groups, while other orthogonal groups build ² D _n and in both cases this refers to a combination of an automorphism of a diagram with an automorphism of a field. It also gives additional exotic Lie groups ² E ₆ and ³ D ₄ ; the latter is defined only over fields with an automorphism of order 3.

With a positive characteristic, additional characteristics give the Suzuki Group - Ri , ² B ₂ , ² F ₄ and ² G ₂ .

Convolution

Coxeter finite group convolutions.

Convolutions of Coxeter affine groups with three naming conventions: the former is the original extended set, the latter is used in the context of , the latter was used by Victor Gershevich Katz for .

A (single-line) Dynkin diagram (finite or affine ), having symmetry (satisfying one condition below), can be minimized by symmetry, which gives a new, in the general case multi-thread (with multiple edges) diagram using a process called convolution . At the level of Lie algebras, this corresponds to taking an invariant subalgebra with an external automorphism group, and the process can be defined purely on the root system without the use of diagrams ^[4] . Further, any multi-thread diagram (finite or infinite) can be obtained by convolution of a single-thread diagram ^[5] .

There is a condition for convolution automorphism so that automorphism is possible - different nodes of the graph in the same orbit (with automorphism) should not be connected by an edge. At the level of the root system, the roots in the same orbit must be orthogonal ^[5] . At the level of the diagrams, this is necessary, because otherwise the resulting diagram will have a loop, since this will combine two nodes having an edge between them, and loops in Dynkin diagrams are not allowed.

The nodes and edges of the resulting (“collapsed”) diagrams are the orbits of the nodes and edges of the original diagrams. The edges are single (not multiple) if adjacent edges do not map to the same edge (especially for valency nodes greater than 2 - “branch points”), otherwise the weight is the number of adjacent edges, and the arrow is directed to the node to which they are incident - “The branch point is mapped to a heterogeneous point.” For example, in D _4, when convolving in G _{2, the} edges in G _{2 are} directed from the external nodes of class 3 (valency 1) to the central nodes (valency 3).

Convolution of the final diagrams ^[6] ^{[note 3]} :

$A_{2n-1}\to C_{n}$ ${\ displaystyle A_ {2n-1} \ to C_ {n}}$

(The automorphism A _{2 n} does not create a convolution, since the middle two nodes are connected by an edge, but are not in the same orbit.)

$D_{n+1}\to B_{n}$ ${\ displaystyle D_ {n + 1} \ to B_ {n}}$
$D_{4}\to G_{2}$ ${\ displaystyle D_ {4} \ to G_ {2}}$ (if you carry out the convolution of the full group or 3-cycle, in addition, $D_{4}\to B_{3}$ ${\ displaystyle D_ {4} \ to B_ {3}}$ in three different ways, if we convolution by involution (an element with order 2))
$E_{6}\to F_{4}$ ${\ displaystyle E_ {6} \ to F_ {4}}$

Similar convolutions exist for affine diagrams:

${\tilde {A}}_{2n-1}\to {\tilde {C}}_{n}$ {\ displaystyle {\ tilde {A}} _ {2n-1} \ to {\ tilde {C}} _ {n}}
${\tilde {D}}_{n+1}\to {\tilde {B}}_{n}$ ${\ displaystyle {\ tilde {D}} _ {n + 1} \ to {\ tilde {B}} _ {n}}$
${\tilde {D}}_{4}\to {\tilde {G}}_{2}$ ${\ displaystyle {\ tilde {D}} _ {4} \ to {\ tilde {G}} _ {2}}$
${\tilde {E}}_{6}\to {\tilde {F}}_{4}$ ${\ displaystyle {\ tilde {E}} _ {6} \ to {\ tilde {F}} _ {4}}$

The convolution record can also be used for Coxeter – Dynkin diagrams ^[7] . We can generalize the permissible convolution of the Dynkin diagram to H _n and I ₂ ( p ). Geometrically, this corresponds to the projections of . You can see that any one-line Dynkin diagram can be folded into I ₂ ( h ), where h is the Coxeter number , geometrically corresponding to the projection onto .

The convolution can be used to reduce questions about (semisimple) Lie algebras to questions about single-stranded algebras together with an automorphism, which can be simpler than considering directly Lie algebras with multiple edges. This can be done by constructing semisimple Lie algebras, for example. See Math Overflow: Folding by Automorphisms for further discussion.

Other chart displays

Root system A ₂	Root system G ₂

Some additional chart displays have a meaningful interpretation, as explained below. However, not all mappings of root systems appear as mappings of diagrams ^[8] .

For example, there are two occurrences of the root systems A ₂ in G ₂ , either as six long roots, or as six short roots. However, the nodes in the diagram G ₂ correspond to one long and one short root, while the nodes in the diagram A ₂ correspond to roots of equal length, and thus this mapping of root systems cannot be expressed as a mapping of diagrams.

Some inclusions of root systems can be expressed as the ratio of graphs when one diagram is generated by a subgraph of another, which means the occurrence of “a subset of nodes along with all edges between them”. This is because removing a node from the Dynkin diagram corresponds to removing a simple root from the root system, which gives the root system with a rank one less. In contrast, removing an edge (or changing the multiplicity of an edge) while maintaining the nodes corresponds to a change in the angles between the roots, which cannot be done without changing the entire root system. Thus, it is possible to meaningfully remove nodes, but not edges. Removing a node from a connected diagram can give a connected diagram (simple Lie algebra) if the node is a leaf, or a disconnected diagram (semisimple but not simple Lie group) with two or three components (the latter for D _n and E _n ). At the level of Lie algebras, these inclusions correspond to Lie subalgebras.

Maximum subgraphs (here “conjugation” means “by means of an automorphism of a diagram ”):

A _{n +1} : A _n , in two conjugate ways.
B _{n +1} : A _n , B _n .
C _{n +1} : A _n , C _n .
D _{n +1} : A _n (two conjugates), D _n .
E _{n + 1} : A _n , D _n , E _n .
- For E ₆ , two of which coincide: $\mathrm {D} _{5}\cong \mathrm {E} _{5}$ ${\ displaystyle \ mathrm {D} _ {5} \ cong \ mathrm {E} _ {5}}$ and are conjugate.
F ₄ : B ₃ , C ₃ .
G ₂ : A ₁ , in two non-conjugate paths (like long roots or short roots).

Finally, the duality of the diagrams corresponds to a change in the direction of the arrows, if any: ^[8] B _n and C _{n are} dual, while F ₄ and G _{2 are} self-dual, since they are single-strand ADE diagrams.

Single Thread Charts

Dynkin's single-line diagrams classify diverse mathematical objects, and it is called the ADE classification .

Dynkin diagrams without multiple edges are called single-stranded . These include charts. $A_{n},D_{n},E_{n}$ ${\ displaystyle A_ {n}, D_ {n}, E_ {n}}$ and the classification of objects by such diagrams is called the ADE classification . In this case, the Dynkin diagrams coincide exactly with the Coxeter diagrams.

Satake Charts

Dynkin diagrams classify complex semisimple Lie algebras. Real semisimple Lie algebras can be classified as complex semisimple Lie algebras, and they are classified by , which can be obtained from Dynkin diagrams by marking some nodes in black (inside the circle) and connecting some other nodes in pairs with arrows by some rules.

History

Evgeny Borisovich Dynkin .

Dynkin diagrams are named after Yevgeny Borisovich Dynkin , who used them in two articles (1946, 1947) to simplify the classification of semisimple Lie algebras ^[9] , see ( E. B. Dynkin 2000 ). After Dynkin left the Soviet Union in 1976, which at that time was regarded as a betrayal, Soviet mathematicians used the name “simple root diagrams” instead of using the author’s surname to refer to the diagrams.

Undirected graphs were previously used by Coxeter (1934) to classify , and the nodes in them corresponded to simple reflections. The graphs were then used by Witt (with length information) (in 1941) in the context of root systems, where the nodes correspond to simple roots, as is used today ^[9] ^[10] . Dynkin then used charts in 1946 and 1947, thanking Coxeter and Witt in a 1947 article.

Agreements

Dynkin diagrams are drawn in many ways ^[10] . The conventions used in this article are generally recognized, with angles of 180 ° for valency nodes 2, 120 ° for valency nodes 3 for D _n and angles of 90 ° / 90 ° / 180 ° valency 3 for E _n , indicating multiplicity with 1, 2 or 3 parallel edges, and indicating the length of the root by indicating the orientation of the edges. In addition to simplicity, these conventions make it possible to show automorphisms of diagrams using Euclidean isometric diagrams.

Alternative conventions include specifying the number of edges for the multiplicity (commonly used in Coxeter diagrams), using color to indicate the length of the root, or using angles of 120 ° for valency 2 nodes to make the nodes more distinguishable.

There are also node numbering conventions. A generally accepted agreement was developed and illustrated in the 1960s in Bourbaki's book ^[11] ^[10] .

Rank 2 Dynkin Charts

Dynkin diagrams are equivalent to generalized Cartan matrices , as shown in the table of Dynkin diagrams of rank 2 by indicating their corresponding 2 x 2 Cartan matrices.

For rank 2, the Cartan matrix has the form:

A=\left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]

{\ displaystyle A = \ left [{\ begin {matrix} 2 & a_ {12} \\ a_ {21} & 2 \ end {matrix}} \ right]}

A multi-edge diagram corresponds to an off-diagonal Cartan matrix with elements -a ₂₁ , -a ₁₂ , where the number of edges of the diagram is max (-a ₂₁ , -a ₁₂ ), and the arrow is directed towards non-unit elements.

The generalized Cartan matrix is a square matrix $A=(a_{ij})$ ${\ displaystyle A = (a_ {ij})}$ such that:

For diagonal elements $a_{ii}=2$ ${\ displaystyle a_ {ii} = 2}$ .
For off-diagonal elements $a_{ij}\leq 0$ ${\ displaystyle a_ {ij} \ leq 0}$ .
$a_{ij}=0$ ${\ displaystyle a_ {ij} = 0}$ if and only if $a_{ji}=0$ ${\ displaystyle a_ {ji} = 0}$

The Cartan matrix determines whether a group is of finite type (if it is positive definite , that is, all eigenvalues are positive), affine type (if the matrix is not positive definite, but positively semidefinite, that is, all eigenvalues are non-negative), or an indefinite type . An indefinite type is often divided into subtypes, for example, the Coxeter group is Lorentzian if it has one negative eigenvalue and all other values are positive. Further, some sources speak of Coxeter hyperbolic groups, but there are several nonequivalent definitions for this concept. In the discussion below, Coxeter hyperbolic groups are understood as a special case of Lorentz groups satisfying additional conditions. Note that for rank 2, all Cartan matrices with a negative determinant correspond to Coxeter hyperbolic groups. But in the general case, most matrices with a negative determinant are neither hyperbolic nor Lorentzian.

The final branches have (-a ₂₁ , -a ₁₂ ) = (1,1), (2,1), (3,1), and affine (with zero determinant) have (-a ₂₁ , -a ₁₂ ) = ( 2.2) or (4.1).

Rank 2 Dynkin Charts
Title groups	Dynkin diagram			Cartan Matrix		Order symmetry	Related single strand group ³
Title groups	(Standard) multi-ribbed graph	Count s values ¹	Graph Coxeter ²	$\left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]$ ${\ displaystyle \ left [{\ begin {matrix} 2 & a_ {12} \\ a_ {21} & 2 \ end {matrix}} \ right]}$	Define (4-a ₂₁ * a ₁₂ )	Order symmetry	Related single strand group ³
Finite (Qualifier> 0)
A ₁ xA ₁				$\left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & 0 \\ 0 & 2 \ end {smallmatrix}} \ right]}$	four	2
A ₂ (neor. ^{[Note 4]} )				$\left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 1 & 2 \ end {smallmatrix}} \ right]}$	3	3
B ₂				$\left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -2 \\ - 1 & 2 \ end {smallmatrix}} \ right]}$	2	four	${A}_{3}$ ${\ displaystyle {A} _ {3}}$
C ₂				$\left[{\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 2 & 2 \ end {smallmatrix}} \ right]}$	2	four	${A}_{3}$ ${\ displaystyle {A} _ {3}}$
BC ₂ (inor.)				$\left[{\begin{smallmatrix}2&-{\sqrt {2}}\\-{\sqrt {2}}&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & - {\ sqrt {2}} \\ - {\ sqrt {2}} & 2 \ end {smallmatrix}} \ right]}$	2	four
G ₂				$\left[{\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 3 & 2 \ end {smallmatrix}} \ right]}$	one	6	${D}_{4}$ ${\ displaystyle {D} _ {4}}$
G ₂ (neor.)				$\left[{\begin{smallmatrix}2&-{\sqrt {3}}\\-{\sqrt {3}}&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & - {\ sqrt {3}} \\ - {\ sqrt {3}} & 2 \ end {smallmatrix}} \ right]}$	one	6
Affine (Qualifier = 0)
A ₁ ⁽¹⁾				$\left[{\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -2 \\ - 2 & 2 \ end {smallmatrix}} \ right]}$	0	∞	${\tilde {A}}_{3}$ ${\ displaystyle {\ tilde {A}} _ {3}}$
A ₂ ⁽²⁾				$\left[{\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 4 & 2 \ end {smallmatrix}} \ right]}$	0	∞	${\tilde {D}}_{4}$ ${\ displaystyle {\ tilde {D}} _ {4}}$
Hyperbolic (Qualifier <0)
				$\left[{\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 5 & 2 \ end {smallmatrix}} \ right]}$	-one	-
				$\left[{\begin{smallmatrix}2&-2\\-3&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -2 \\ - 3 & 2 \ end {smallmatrix}} \ right]}$	-2	-
				$\left[{\begin{smallmatrix}2&-1\\-6&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 6 & 2 \ end {smallmatrix}} \ right]}$	-2	-
				$\left[{\begin{smallmatrix}2&-1\\-7&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 7 & 2 \ end {smallmatrix}} \ right]}$	-3	-
				$\left[{\begin{smallmatrix}2&-2\\-4&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -2 \\ - 4 & 2 \ end {smallmatrix}} \ right]}$	-four	-
				$\left[{\begin{smallmatrix}2&-1\\-8&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -1 \\ - 8 & 2 \ end {smallmatrix}} \ right]}$	-four	-
				$\left[{\begin{smallmatrix}2&-3\\-3&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -3 \\ - 3 & 2 \ end {smallmatrix}} \ right]}$	-five	-
				$\left[{\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}}\right]$ ${\ displaystyle \ left [{\ begin {smallmatrix} 2 & -b \\ - a & 2 \ end {smallmatrix}} \ right]}$	4-ab <0	-
Note ¹ : For hyperbolic groups, (a ₁₂ * a ₂₁ > 4), the multi-edge style is not used, and the values (a ₂₁ , a ₁₂ ) are indicated directly on the edge. This is not commonly used for finite and affinity groups ^[12] . Note ² : For undirected groups, Dynkin diagrams and Coxeter diagrams are equivalent. The edges in them are usually marked with their order of symmetry, and the edges of order 3 are not marked. Note ³ : Many multi-ribbed groups can be obtained from single-stranded groups of higher rank using a suitable convolution operation .

Dynkin Finite Charts

Finite Dynkin graphs with the number of nodes from 1 to 9
Rank					Exceptional Lee groups
Rank	${A}_{1+}$ ${\ displaystyle {A} _ {1+}}$	${B}_{2+}$ ${\ displaystyle {B} _ {2+}}$	${C}_{2+}$ ${\ displaystyle {C} _ {2+}}$	${D}_{2+}$ ${\ displaystyle {D} _ {2+}}$	${E}_{3-8}$	${G}_{2}$ {\ displaystyle {G} _ {2}} / ${F}_{4}$ ${\ displaystyle {F} _ {4}}$
one	A ₁
2	A ₂	B ₂	C ₂ = B ₂	D ₂ = A ₁ xA ₁		G ₂
3	A ₃	B ₃	C ₃	D ₃ = A ₃	E ₃ = A ₂ xA ₁
four	A ₄	B ₄	C ₄	D ₄	E ₄ = A ₄	F ₄
five	A ₅	B ₅	C ₅	D ₅	E ₅ = D ₅
6	A ₆	B ₆	C ₆	D ₆	E ₆
7	A ₇	B ₇	C ₇	D ₇	E ₇
eight	A ₈	B ₈	C ₈	D ₈	E ₈
9	A ₉	B ₉	C ₉	D ₉
10+	..	..	..	..

Dynkin diagrams

There are extensions to Dynkin diagrams, namely , Dynkin affine diagrams . These diagrams classify the Cartan matrices of . The classification was carried out in the article by Katz ^[13] , the list is given in the same article on pages 53-55. Affine diagrams are denoted as $X_{l}^{(1)},X_{l}^{(2)}$ ${\ displaystyle X_ {l} ^ {(1)}, X_ {l} ^ {(2)}}$ or $X_{l}^{(3)},$ ${\ displaystyle X_ {l} ^ {(3)},}$ where X is the letter of the corresponding final diagram, and the superscript indicates the series of affine diagrams into which the diagrams belong. The first of the series, $X_{l}^{(1)},$ ${\ displaystyle X_ {l} ^ {(1)},}$ best known, called extended Dynkin diagrams and is marked with a tilde (~), and sometimes a + sign in the upper index ^[14] , for example, ${\tilde {A}}_{5}=A_{5}^{(1)}=A_{5}^{+}$ ${\ displaystyle {\ tilde {A}} _ {5} = A_ {5} ^ {(1)} = A_ {5} ^ {+}}$ . Series (2) and (3) are called twisted affine diagrams .

See Dynkin Chart Generator for charts.

Many advanced Dynkin affinity diagrams with added nodes (marked in green) ( $n\geq 3$ ${\ displaystyle n \ geq 3}$ for $B_{n}$ ${\ displaystyle B_ {n}}$ and $n\geq 4$ ${\ displaystyle n \ geq 4}$ for $D_{n}$ ${\ displaystyle D_ {n}}$ )	Twisted affinity diagrams are labeled (2) or (3) in the superscript. ( k is equal to the number of yellow nodes of the graph)

The table below shows all Dynkin graphs for affine groups of up to 10 nodes. Extended Dynkin graphs are indicated as families with ~ and correspond to the finite graphs above with one added node. Other oriented graphs are given with superscripts (2) or (3) and they are convolutions of higher order groups. They fall into the category of Twisted affine diagrams ^[15] .

Connected affine graphs of Dynkin with the number of nodes from 2 to 10
(grouped as undirected graphs)
Rank	${\tilde {A}}_{1+}$ ${\ displaystyle {\ tilde {A}} _ {1+}}$	${\tilde {B}}_{3+}$ ${\ displaystyle {\ tilde {B}} _ {3+}}$	${\tilde {C}}_{2+}$ ${\ displaystyle {\ tilde {C}} _ {2+}}$	${\tilde {D}}_{4+}$ ${\ displaystyle {\ tilde {D}} _ {4+}}$	E / f / g
2	${\tilde {A}}_{1}$ ${\ displaystyle {\ tilde {A}} _ {1}}$ or ${A}_{1}^{(1)}$ ${\ displaystyle {A} _ {1} ^ {(1)}}$		${A}_{2}^{(2)}$ ${\ displaystyle {A} _ {2} ^ {(2)}}$ :
3	${\tilde {A}}_{2}$ ${\ displaystyle {\ tilde {A}} _ {2}}$ or ${A}_{2}^{(1)}$ ${\ displaystyle {A} _ {2} ^ {(1)}}$ (cm.)		${\tilde {C}}_{2}$ ${\ displaystyle {\ tilde {C}} _ {2}}$ or ${C}_{2}^{(1)}$ ${\ displaystyle {C} _ {2} ^ {(1)}}$ (cm.) ${D}_{5}^{(2)}$ ${\ displaystyle {D} _ {5} ^ {(2)}}$ : ${A}_{4}^{(2)}$ ${\ displaystyle {A} _ {4} ^ {(2)}}$ :		${\tilde {G}}_{2}$ ${\ displaystyle {\ tilde {G}} _ {2}}$ or ${G}_{2}^{(1)}$ ${\ displaystyle {G} _ {2} ^ {(1)}}$ (cm.) ${D}_{4}^{(3)}$ ${\ displaystyle {D} _ {4} ^ {(3)}}$
four	${\tilde {A}}_{3}$ ${\ displaystyle {\ tilde {A}} _ {3}}$ or ${A}_{3}^{(1)}$ ${\ displaystyle {A} _ {3} ^ {(1)}}$ (cm.)	${\tilde {B}}_{3}$ ${\ displaystyle {\ tilde {B}} _ {3}}$ or ${B}_{3}^{(1)}$ ${\ displaystyle {B} _ {3} ^ {(1)}}$ (cm.) ${A}_{5}^{(2)}$ ${\ displaystyle {A} _ {5} ^ {(2)}}$ :	${\tilde {C}}_{3}$ ${\ displaystyle {\ tilde {C}} _ {3}}$ or ${C}_{3}^{(1)}$ ${\ displaystyle {C} _ {3} ^ {(1)}}$ (cm.) ${D}_{6}^{(2)}$ ${\ displaystyle {D} _ {6} ^ {(2)}}$ : ${A}_{6}^{(2)}$ ${\ displaystyle {A} _ {6} ^ {(2)}}$ :
five	${\tilde {A}}_{4}$ ${\ displaystyle {\ tilde {A}} _ {4}}$ or ${A}_{4}^{(1)}$ ${\ displaystyle {A} _ {4} ^ {(1)}}$ (cm.)	${\tilde {B}}_{4}$ ${\ displaystyle {\ tilde {B}} _ {4}}$ or ${B}_{4}^{(1)}$ ${\ displaystyle {B} _ {4} ^ {(1)}}$ (cm.) ${A}_{7}^{(2)}$ ${\ displaystyle {A} _ {7} ^ {(2)}}$ :	${\tilde {C}}_{4}$ ${\ displaystyle {\ tilde {C}} _ {4}}$ or ${C}_{4}^{(1)}$ ${\ displaystyle {C} _ {4} ^ {(1)}}$ (cm.) ${D}_{7}^{(2)}$ ${\ displaystyle {D} _ {7} ^ {(2)}}$ : ${A}_{8}^{(2)}$ ${\ displaystyle {A} _ {8} ^ {(2)}}$ :	${\tilde {D}}_{4}$ ${\ displaystyle {\ tilde {D}} _ {4}}$ or ${D}_{4}^{(1)}$ ${\ displaystyle {D} _ {4} ^ {(1)}}$ (cm.)	${\tilde {F}}_{4}$ ${\ displaystyle {\ tilde {F}} _ {4}}$ or ${F}_{4}^{(1)}$ ${\ displaystyle {F} _ {4} ^ {(1)}}$ (cm.) ${E}_{6}^{(2)}$ ${\ displaystyle {E} _ {6} ^ {(2)}}$
6	${\tilde {A}}_{5}$ ${\ displaystyle {\ tilde {A}} _ {5}}$ or ${A}_{5}^{(1)}$ ${\ displaystyle {A} _ {5} ^ {(1)}}$ (cm.)	${\tilde {B}}_{5}$ ${\ displaystyle {\ tilde {B}} _ {5}}$ or ${B}_{5}^{(1)}$ ${\ displaystyle {B} _ {5} ^ {(1)}}$ (cm.) ${A}_{9}^{(2)}$ ${\ displaystyle {A} _ {9} ^ {(2)}}$ :	${\tilde {C}}_{5}$ ${\ displaystyle {\ tilde {C}} _ {5}}$ or ${C}_{5}^{(1)}$ ${\ displaystyle {C} _ {5} ^ {(1)}}$ (cm.) ${D}_{8}^{(2)}$ ${\ displaystyle {D} _ {8} ^ {(2)}}$ : ${A}_{10}^{(2)}$ ${\ displaystyle {A} _ {10} ^ {(2)}}$ :	${\tilde {D}}_{5}$ ${\ displaystyle {\ tilde {D}} _ {5}}$ or ${D}_{5}^{(1)}$ ${\ displaystyle {D} _ {5} ^ {(1)}}$ (cm.)
7	${\tilde {A}}_{6}$ ${\ displaystyle {\ tilde {A}} _ {6}}$ or ${A}_{6}^{(1)}$ ${\ displaystyle {A} _ {6} ^ {(1)}}$ (cm.)	${\tilde {B}}_{6}$ ${\ displaystyle {\ tilde {B}} _ {6}}$ or ${B}_{6}^{(1)}$ ${\ displaystyle {B} _ {6} ^ {(1)}}$ ${A}_{11}^{(2)}$ ${\ displaystyle {A} _ {11} ^ {(2)}}$ :	${\tilde {C}}_{6}$ ${\ displaystyle {\ tilde {C}} _ {6}}$ or ${C}_{6}^{(1)}$ ${\ displaystyle {C} _ {6} ^ {(1)}}$ ${D}_{9}^{(2)}$ ${\ displaystyle {D} _ {9} ^ {(2)}}$ : ${A}_{12}^{(2)}$ ${\ displaystyle {A} _ {12} ^ {(2)}}$ :	${\tilde {D}}_{6}$ ${\ displaystyle {\ tilde {D}} _ {6}}$ or ${D}_{6}^{(1)}$ ${\ displaystyle {D} _ {6} ^ {(1)}}$	${\tilde {E}}_{6}$ ${\ displaystyle {\ tilde {E}} _ {6}}$ or ${E}_{6}^{(1)}$ ${\ displaystyle {E} _ {6} ^ {(1)}}$
eight	${\tilde {A}}_{7}$ ${\ displaystyle {\ tilde {A}} _ {7}}$ or ${A}_{7}^{(1)}$ ${\ displaystyle {A} _ {7} ^ {(1)}}$ (cm.)	${\tilde {B}}_{7}$ ${\ displaystyle {\ tilde {B}} _ {7}}$ or ${B}_{7}^{(1)}$ ${\ displaystyle {B} _ {7} ^ {(1)}}$ (cm.) ${A}_{13}^{(2)}$ ${\ displaystyle {A} _ {13} ^ {(2)}}$ :	${\tilde {C}}_{7}$ ${\ displaystyle {\ tilde {C}} _ {7}}$ or ${C}_{7}^{(1)}$ ${\ displaystyle {C} _ {7} ^ {(1)}}$ ${D}_{10}^{(2)}$ ${\ displaystyle {D} _ {10} ^ {(2)}}$ : ${A}_{14}^{(2)}$ ${\ displaystyle {A} _ {14} ^ {(2)}}$ :	${\tilde {D}}_{7}$ ${\ displaystyle {\ tilde {D}} _ {7}}$ or ${D}_{7}^{(1)}$ ${\ displaystyle {D} _ {7} ^ {(1)}}$ (cm.)	${\tilde {E}}_{7}$ ${\ displaystyle {\ tilde {E}} _ {7}}$ or ${E}_{7}^{(1)}$ ${\ displaystyle {E} _ {7} ^ {(1)}}$
9	${\tilde {A}}_{8}$ ${\ displaystyle {\ tilde {A}} _ {8}}$ or ${A}_{8}^{(1)}$ ${\ displaystyle {A} _ {8} ^ {(1)}}$ (cm.)	${\tilde {B}}_{8}$ ${\ displaystyle {\ tilde {B}} _ {8}}$ or ${B}_{8}^{(1)}$ ${\ displaystyle {B} _ {8} ^ {(1)}}$ ${A}_{15}^{(2)}$ ${\ displaystyle {A} _ {15} ^ {(2)}}$ :	${\tilde {C}}_{8}$ ${\ displaystyle {\ tilde {C}} _ {8}}$ or ${C}_{8}^{(1)}$ ${\ displaystyle {C} _ {8} ^ {(1)}}$ ${D}_{11}^{(2)}$ ${\ displaystyle {D} _ {11} ^ {(2)}}$ : ${A}_{16}^{(2)}$ ${\ displaystyle {A} _ {16} ^ {(2)}}$ :	${\tilde {D}}_{8}$ ${\ displaystyle {\ tilde {D}} _ {8}}$ or ${D}_{8}^{(1)}$ ${\ displaystyle {D} _ {8} ^ {(1)}}$	${\tilde {E}}_{8}$ ${\ displaystyle {\ tilde {E}} _ {8}}$ or ${E}_{8}^{(1)}$ ${\ displaystyle {E} _ {8} ^ {(1)}}$
ten	${\tilde {A}}_{9}$ ${\ displaystyle {\ tilde {A}} _ {9}}$ or ${A}_{9}^{(1)}$ ${\ displaystyle {A} _ {9} ^ {(1)}}$ (cm.)	${\tilde {B}}_{9}$ ${\ displaystyle {\ tilde {B}} _ {9}}$ or ${B}_{9}^{(1)}$ ${\ displaystyle {B} _ {9} ^ {(1)}}$ ${A}_{17}^{(2)}$ ${\ displaystyle {A} _ {17} ^ {(2)}}$ :	${\tilde {C}}_{9}$ ${\ displaystyle {\ tilde {C}} _ {9}}$ or ${C}_{9}^{(1)}$ ${\ displaystyle {C} _ {9} ^ {(1)}}$ ${D}_{12}^{(2)}$ ${\ displaystyle {D} _ {12} ^ {(2)}}$ : ${A}_{18}^{(2)}$ ${\ displaystyle {A} _ {18} ^ {(2)}}$ :	${\tilde {D}}_{9}$ ${\ displaystyle {\ tilde {D}} _ {9}}$ or ${D}_{9}^{(1)}$ ${\ displaystyle {D} _ {9} ^ {(1)}}$
eleven	...	...	...	...

Dynkin's hyperbolic diagrams and higher levels

The set of compact and non-compact hyperbolic graphs of Dynkin was listed in the article by Carbone et al. ^[16] All rank 3 hyperbolic graphs are compact. Compact hyperbolic Dynkin diagrams exist up to rank 5, and non-compact hyperbolic graphs exist up to rank 10.

Number of charts
Rank	Compact	Non-compact	Total
3	31	93	123
four	3	50	53
five	one	21	22
6	0	22	22
7	0	four	four
eight	0	five	five
9	0	five	five
ten	0	four	four

Compact hyperbolic Dynkin diagrams

Compact hyperbolic graphs
Rank 3		Rank 4	Rank 5
Linear graphs (6 4 2): H ₁₀₀ ⁽³⁾ : H ₁₀₁ ⁽³⁾ : H ₁₀₅ ⁽³⁾ : H ₁₀₆ ⁽³⁾ : (6 6 2): H ₁₁₄ ⁽³⁾ : H ₁₁₅ ⁽³⁾ : H ₁₁₆ ⁽³⁾ :	Cyclic graphs (4 3 3): H ₁ ⁽³⁾ : (4 4 3): 3 forms ... (4 4 4): 2 forms ... (6 3 3): H ₃ ⁽³⁾ : (6 4 3): 4 forms ... (6 4 4): 4 forms ... (6 6 3): 3 forms ... (6 6 4): 4 forms ... (6 6 6): 2 forms ...	(4 3 3 3): H ₈ ⁽⁴⁾ : H ₁₃ ⁽⁴⁾ : (4 3 4 3): H ₁₄ ⁽⁴⁾ :	(4 3 3 3 3): H ₇ ⁽⁵⁾ :

Non-compact (substantially extended forms)

Some notations used in theoretical physics , in areas such as M-theory , use the superscript “+” for extended groups instead of “~”, which makes it possible to define stronger group extensions.

Extended Dynkin diagrams (affine) are given the “+” index and they have one additional node. (Same as "~")
Significantly expanded Dynkin diagrams (hyperbolic) are given the index “^” or “++” and they have two additional nodes.
Heavily expanded Dynkin diagrams with 3 additional nodes are given the index “+++”.

Some examples of substantially extended (hyperbolic) Dynkin diagrams
Rank	${AE}_{n}$ ${\ displaystyle {AE} _ {n}}$ = A _n-2 ^{(1) ^}	${BE}_{n}$ ${\ displaystyle {BE} _ {n}}$ = B _n-2 ^{(1) ^} ${CE}_{n}$ ${\ displaystyle {CE} _ {n}}$	C _n-2 ^{(1) ^}	E / F / G
3	${AE}_{3}$ ${\ displaystyle {AE} _ {3}}$ :
four	${AE}_{4}$ ${\ displaystyle {AE} _ {4}}$ :		C ₂ ^{(1) ^} A ₄ ^{(2) '^} A ₄ ^{(2) ^} D ₃ ^{(2) ^}	G ₂ ^{(1) ^} D ₄ ^{(3) ^}
five	${AE}_{5}$ ${\ displaystyle {AE} _ {5}}$ :	${BE}_{5}$ ${\ displaystyle {BE} _ {5}}$