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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
, a Dynkin diagram, named for
Eugene Dynkin Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially sem ...
, is a type of
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s over
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
s, in the classification of
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
s and other
finite reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...
s, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
, in which case they correspond to
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
s and semi-simple Lie algebras, while in other cases they are assumed to be
undirected In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named.


Classification of semisimple Lie algebras

The fundamental interest in Dynkin diagrams is that they classify
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s over
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
s. One classifies such Lie algebras via their
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
, which can be represented by a Dynkin diagram. One then classifies Dynkin diagrams according to the constraints they must satisfy, as described below. Dropping the direction on the graph edges corresponds to replacing a root system by the
finite reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...
it generates, the so-called
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
, and thus undirected Dynkin diagrams classify Weyl groups. They have the following correspondence for the Lie algebras associated to classical groups over the complex numbers: * A_n: \mathfrak _, the special linear Lie algebra. * B_n: \mathfrak_, the odd-dimensional
special orthogonal Lie algebra In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. * C_n: \mathfrak _, the symplectic Lie algebra. * D_n: \mathfrak_, the even-dimensional
special orthogonal Lie algebra In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
(n>1). For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide.


Related classifications

Dynkin diagrams can be interpreted as classifying many distinct, related objects, and the notation "A''n'', B''n'', ..." is used to refer to ''all'' such interpretations, depending on context; this ambiguity can be confusing. The central classification is that a simple Lie algebra has a root system, to which is associated an (oriented) Dynkin diagram; all three of these may be referred to as B''n'', for instance. The ''un''oriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, which is the
finite reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...
associated to the root system. Thus B''n'' may refer to the unoriented diagram (a special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxeter group. Although the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots. Likewise, while Dynkin diagram notation is standardized, Coxeter diagram and group notation is varied and sometimes agrees with Dynkin diagram notation and sometimes does not. Lastly, ''sometimes'' associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include: * The
root lattice In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
generated by the root system, as in the E8 lattice. This is naturally defined, but not one-to-one – for example, A2 and G2 both generate the
hexagonal lattice The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120� ...
. * An associated polytope – for example Gosset 421 polytope may be referred to as "the E8 polytope", as its vertices are derived from the E8 root system and it has the E8 Coxeter group as symmetry group. * An associated quadratic form or manifold – for example, the E8 manifold has intersection form given by the E8 lattice. These latter notations are mostly used for objects associated with exceptional diagrams – objects associated to the regular diagrams (A, B, C, D) instead have traditional names. The index (the ''n'') equals to the number of nodes in the diagram, the number of simple roots in a basis, the dimension of the root lattice and span of the root system, the number of generators of the Coxeter group, and the rank of the Lie algebra. However, ''n'' does not equal the dimension of the defining module (a
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, B_4 corresponds to \mathfrak_ = \mathfrak_9, which naturally acts on 9-dimensional space, but has rank 4 as a Lie algebra. The simply laced Dynkin diagrams, those with no multiple edges (A, D, E) classify many further mathematical objects; see discussion at
ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
.


Example: A2

For example, the symbol A_2 may refer to: * The Dynkin diagram with 2 connected nodes, , which may also be interpreted as a
Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
. * The
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
with 2 simple roots at a 2\pi/3 (120 degree) angle. * The Lie algebra \mathfrak_ = \mathfrak_3 of
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
2. * The
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
of symmetries of the roots (reflections in the hyperplane orthogonal to the roots), isomorphic to the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
S_3 (of order 6). * The abstract
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
, presented by generators and relations, \left\langle r_1,r_2 \mid (r_1)^2=(r_2)^2=(r_ir_j)^3=1\right\rangle.


Construction from root systems

Consider a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
, assumed to be reduced and integral (or "crystallographic"). In many applications, this root system will arise from a
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
. Let \Delta be a set of positive simple roots. We then construct a diagram from \Delta as follows. Form a graph with one vertex for each element of \Delta. Then insert edges between each pair of vertices according to the following recipe. If the roots corresponding to the two vertices are orthogonal, there is no edge between the vertices. If the angle between the two roots is 120 degrees, we put one edge between the vertices. If the angle is 135 degrees, we put two edges, and if the angle is 150 degrees, we put three edges. (These four cases exhaust all possible angles between pairs of positive simple roots.) Finally, if there are any edges between a given pair of vertices, we decorate them with an arrow pointing from the vertex corresponding to the longer root to the vertex corresponding to the shorter one. (The arrow is omitted if the roots have the same length.) Thinking of the arrow as a "greater than" sign makes it clear which way the arrow should go. Dynkin diagrams lead to a
classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...
of root systems. The angles and length ratios between roots are
related ''Related'' is an American comedy-drama television series that aired on The WB from October 5, 2005, to March 20, 2006. It revolves around the lives of four close-knit sisters of Italian descent, raised in Brooklyn and living in Manhattan. The ...
. Thus, the edges for non-orthogonal roots may alternatively be described as one edge for a length ratio of 1, two edges for a length ratio of \sqrt, and three edges for a length ratio of \sqrt. (There are no edges when the roots are orthogonal, regardless of the length ratio.) In the A_2 root system, shown at right, the roots labeled \alpha and \beta form a base. Since these two roots are at angle of 120 degrees (with a length ratio of 1), the Dynkin diagram consists of two vertices connected by a single edge: .


Constraints

Dynkin diagrams must satisfy certain constraints; these are essentially those satisfied by finite
Coxeter–Dynkin diagram In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...
s, together with an additional crystallographic constraint.


Connection with Coxeter diagrams

Dynkin diagrams are closely related to Coxeter diagrams of finite
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s, and the terminology is often conflated.In this section we refer to the general class as "Coxeter diagrams" rather than "Coxeter–Dynkin diagrams" for clarity, as there is great potential for confusion, and for concision. Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects: ;Partly directed: Dynkin diagrams are ''partly
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
'' – any multiple edge (in Coxeter terms, labeled with "4" or above) has a direction (an arrow pointing from one node to the other); thus Dynkin diagrams have ''more'' data than the underlying Coxeter diagram (undirected graph). :At the level of root systems the direction corresponds to pointing towards the shorter vector; edges labeled "3" have no direction because the corresponding vectors must have equal length. (Caution: Some authors reverse this convention, with the arrow pointing towards the longer vector.) ;Crystallographic restriction: Dynkin diagrams must satisfy an additional restriction, namely that the only allowable edge labels are 2, 3, 4, and 6, a restriction not shared by Coxeter diagrams, so not every Coxeter diagram of a finite group comes from a Dynkin diagram. :At the level of root systems this corresponds to the crystallographic restriction theorem, as the roots form a lattice. A further difference, which is only stylistic, is that Dynkin diagrams are conventionally drawn with double or triple edges between nodes (for ''p'' = 4, 6), rather than an edge labeled with "''p''". The term "Dynkin diagram" at times refers to the ''directed'' graph, at times to the ''undirected'' graph. For precision, in this article "Dynkin diagram" will mean ''directed,'' and the underlying undirected graph will be called an "undirected Dynkin diagram". Then Dynkin diagrams and Coxeter diagrams may be related as follows: By this is meant that Coxeter diagrams of finite groups correspond to
point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
s generated by reflections, while Dynkin diagrams must satisfy an additional restriction corresponding to the crystallographic restriction theorem, and that Coxeter diagrams are undirected, while Dynkin diagrams are (partly) directed. The corresponding mathematical objects classified by the diagrams are: The blank in the upper right, corresponding to directed graphs with underlying undirected graph any Coxeter diagram (of a finite group), can be defined formally, but is little-discussed, and does not appear to admit a simple interpretation in terms of mathematical objects of interest. There are natural maps down – from Dynkin diagrams to undirected Dynkin diagrams; respectively, from root systems to the associated Weyl groups – and right – from undirected Dynkin diagrams to Coxeter diagrams; respectively from Weyl groups to finite Coxeter groups. The down map is onto (by definition) but not one-to-one, as the ''B''''n'' and ''C''''n'' diagrams map to the same undirected diagram, with the resulting Coxeter diagram and Weyl group thus sometimes denoted ''BC''''n''. The right map is simply an inclusion – undirected Dynkin diagrams are special cases of Coxeter diagrams, and Weyl groups are special cases of finite Coxeter groups – and is not onto, as not every Coxeter diagram is an undirected Dynkin diagram (the missed diagrams being ''H''3, ''H''4 and ''I''2(''p'') for ''p'' = 5 ''p'' ≥ 7), and correspondingly not every finite Coxeter group is a Weyl group.


Isomorphisms

Dynkin diagrams are conventionally numbered so that the list is non-redundant: n \geq 1 for A_n, n \geq 2 for B_n, n \geq 3 for C_n, n \geq 4 for D_n, and E_n starting at n=6. The families can however be defined for lower ''n,'' yielding
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
s of diagrams, and corresponding exceptional isomorphisms of Lie algebras and associated Lie groups. Trivially, one can start the families at n=0 or n=1, which are all then isomorphic as there is a unique empty diagram and a unique 1-node diagram. The other isomorphisms of connected Dynkin diagrams are: * A_1 \cong B_1 \cong C_1 * B_2 \cong C_2 * D_2 \cong A_1 \times A_1 * D_3 \cong A_3 * E_3 \cong A_1 \times A_2 * E_4 \cong A_4 * E_5 \cong D_5 These isomorphisms correspond to isomorphism of simple and semisimple Lie algebras, which also correspond to certain isomorphisms of Lie group forms of these. They also add context to the En family.


Automorphisms

In addition to isomorphism between different diagrams, some diagrams also have self-isomorphisms or "
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s". Diagram automorphisms correspond to
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
s of the Lie algebra, meaning that the outer automorphism group Out = Aut/Inn equals the group of diagram automorphisms.Outer automorphisms of simple Lie Algebras
/ref> The diagrams that have non-trivial automorphisms are A''n'' (n > 1), D''n'' (n > 1), and E6. In all these cases except for D4, there is a single non-trivial automorphism (Out = ''C''2, the cyclic group of order 2), while for D4, the automorphism group is the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on three letters (''S''3, order 6) – this phenomenon is known as " triality". It happens that all these diagram automorphisms can be realized as Euclidean symmetries of how the diagrams are conventionally drawn in the plane, but this is just an artifact of how they are drawn, and not intrinsic structure. For A''n'', the diagram automorphism is reversing the diagram, which is a line. The nodes of the diagram index the fundamental weights, which (for A''n''−1) are \bigwedge^i C^n for i=1,\dots,n, and the diagram automorphism corresponds to the duality \bigwedge^i C^n \mapsto \bigwedge^ C^n. Realized as the Lie algebra \mathfrak_, the outer automorphism can be expressed as negative transpose, T \mapsto -T^, which is how the dual representation acts. For D''n'', the diagram automorphism is switching the two nodes at the end of the Y, and corresponds to switching the two
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
s. Realized as the Lie algebra \mathfrak_, the outer automorphism can be expressed as conjugation by a matrix in O(2''n'') with determinant −1. When ''n'' = 3, one has \mathrm_3 \cong \mathrm_3, so their automorphisms agree, while \mathrm_2 \cong \mathrm_1 \times \mathrm_1 is disconnected, and the automorphism corresponds to switching the two nodes. For D4, the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
is isomorphic to the two spin representations, and the resulting
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on three letter (''S''3, or alternatively the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of order 6, Dih3) corresponds both to automorphisms of the Lie algebra and automorphisms of the diagram. The automorphism group of E6 corresponds to reversing the diagram, and can be expressed using Jordan algebras. Disconnected diagrams, which correspond to ''semi''simple Lie algebras, may have automorphisms from exchanging components of the diagram. In positive characteristic there are additional "diagram automorphisms" – roughly speaking, in characteristic ''p'' one is sometimes allowed to ignore the arrow on bonds of multiplicity ''p'' in the Dynkin diagram when taking diagram automorphisms. Thus in characteristic 2 there is an order 2 automorphism of \mathrm_2 \cong \mathrm_2 and of F4, while in characteristic 3 there is an order 2 automorphism of G2. But doesn't apply in all circumstances: for example, such automorphisms need not arise as automorphisms of the corresponding algebraic group, but rather on the level of points valued in a finite field.


Construction of Lie groups via diagram automorphisms

Diagram automorphisms in turn yield additional Lie groups and
groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...
, which are of central importance in the classification of finite simple groups. The
Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The ...
construction of Lie groups in terms of their Dynkin diagram does not yield some of the classical groups, namely the unitary groups and the non- split orthogonal groups. The Steinberg groups construct the unitary groups 2A''n'', while the other orthogonal groups are constructed as 2D''n'', where in both cases this refers to combining a diagram automorphism with a field automorphism. This also yields additional exotic Lie groups 2E6 and 3D4, the latter only defined over fields with an order 3 automorphism. The additional diagram automorphisms in positive characteristic yield the Suzuki–Ree groups, 2B2, 2F4, and 2G2.


Folding

A (simply-laced) Dynkin diagram (finite or
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
) that has a symmetry (satisfying one condition, below) can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called folding (due to most symmetries being 2-fold). At the level of Lie algebras, this corresponds to taking the invariant subalgebra under the outer automorphism group, and the process can be defined purely with reference to root systems, without using diagrams. Further, every multiply laced diagram (finite or infinite) can be obtained by folding a simply-laced diagram.Folding by Automorphisms
, John Stembridge, 4pp., 79K, 20 August 2008

/ref> The one condition on the automorphism for folding to be possible is that distinct nodes of the graph in the same orbit (under the automorphism) must not be connected by an edge; at the level of root systems, roots in the same orbit must be orthogonal. At the level of diagrams, this is necessary as otherwise the quotient diagram will have a loop, due to identifying two nodes but having an edge between them, and loops are not allowed in Dynkin diagrams. The nodes and edges of the quotient ("folded") diagram are the orbits of nodes and edges of the original diagram; the edges are single unless two incident edges map to the same edge (notably at nodes of valence greater than 2) – a "branch point" of the map, in which case the weight is the number of incident edges, and the arrow points ''towards'' the node at which they are incident – "the branch point maps to the non-homogeneous point". For example, in D4 folding to G2, the edge in G2 points from the class of the 3 outer nodes (valence 1), to the class of the central node (valence 3). The foldings of finite diagrams are:Note that Stekloshchik uses an arrow convention opposite to that of this article. * A_ \to C_n :(The automorphism of A2''n'' does not yield a folding because the middle two nodes are connected by an edge, but in the same orbit.) * D_ \to B_n * D_4 \to G_2 (if quotienting by the full group or a 3-cycle, in addition to D_4 \to B_3 in 3 different ways, if quotienting by an involution) * E_6 \to F_4 Similar foldings exist for affine diagrams, including: * \tilde A_ \to \tilde C_n * \tilde D_ \to \tilde B_n * \tilde D_4 \to \tilde G_2 * \tilde E_6 \to \tilde F_4 The notion of foldings can also be applied more generally to
Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
s – notably, one can generalize allowable quotients of Dynkin diagrams to Hn and I2(''p''). Geometrically this corresponds to projections of
uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimen ...
s. Notably, any simply laced Dynkin diagram can be folded to I2(''h''), where ''h'' is the
Coxeter number In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ...
, which corresponds geometrically to projection to the
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
. Folding can be applied to reduce questions about (semisimple) Lie algebras to questions about simply-laced ones, together with an automorphism, which may be simpler than treating multiply laced algebras directly; this can be done in constructing the semisimple Lie algebras, for instance. Se
Math Overflow: Folding by Automorphisms
for further discussion.


Other maps of diagrams

Some additional maps of diagrams have meaningful interpretations, as detailed below. However, not all maps of root systems arise as maps of diagrams. For example, there are two inclusions of root systems of A2 in G2, either as the six long roots or the six short roots. However, the nodes in the G2 diagram correspond to one long root and one short root, while the nodes in the A2 diagram correspond to roots of equal length, and thus this map of root systems cannot be expressed as a map of the diagrams. Some inclusions of root systems can be expressed as one diagram being an
induced subgraph In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Defini ...
of another, meaning "a subset of the nodes, with all edges between them". This is because eliminating a node from a Dynkin diagram corresponds to removing a simple root from a root system, which yields a root system of rank one lower. By contrast, removing an edge (or changing the multiplicity of an edge) while leaving the nodes unchanged corresponds to changing the angles between roots, which cannot be done without changing the entire root system. Thus, one can meaningfully remove nodes, but not edges. Removing a node from a connected diagram may yield a connected diagram (simple Lie algebra), if the node is a leaf, or a disconnected diagram (semisimple but not simple Lie algebra), with either two or three components (the latter for D''n'' and E''n''). At the level of Lie algebras, these inclusions correspond to sub-Lie algebras. The maximal subgraphs are as follows; subgraphs related by a diagram automorphism are labeled "conjugate": * A''n''+1: A''n'', in 2 conjugate ways. * B''n''+1: A''n'', B''n''. * C''n''+1: A''n'', C''n''. * D''n''+1: A''n'' (2 conjugate ways), D''n''. * E''n''+1: A''n'', D''n'', E''n''. ** For E6, two of these coincide: \mathrm_5 \cong \mathrm_5 and are conjugate. * F4: B3, C3. * G2: A1, in 2 non-conjugate ways (as a long root or a short root). Finally, duality of diagrams corresponds to reversing the direction of arrows, if any: Bn and Cn are dual, while F4, and G2 are self-dual, as are the simply-laced ADE diagrams.


Simply laced

A Dynkin diagram with no multiple edges is called simply laced, as are the corresponding Lie algebra and Lie group. These are the A_n, D_n, E_n diagrams, and phenomena that such diagrams classify are referred to as an
ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
. In this case the Dynkin diagrams exactly coincide with Coxeter diagrams, as there are no multiple edges.


Satake diagrams

Dynkin diagrams classify ''complex'' semisimple Lie algebras. Real semisimple Lie algebras can be classified as real forms of complex semisimple Lie algebras, and these are classified by
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
s, which are obtained from the Dynkin diagram by labeling some vertices black (filled), and connecting some other vertices in pairs by arrows, according to certain rules.


History

Dynkin diagrams are named for
Eugene Dynkin Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially sem ...
, who used them in two papers (1946, 1947) simplifying the classification of semisimple Lie algebras; see . When Dynkin left the Soviet Union in 1976, which was at the time considered tantamount to treason, Soviet mathematicians were directed to refer to "diagrams of simple roots" rather than use his name. Undirected graphs had been used earlier by Coxeter (1934) to classify
reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent c ...
s, where the nodes corresponded to simple reflections; the graphs were then used (with length information) by Witt (1941) in reference to root systems, with the nodes corresponding to simple roots, as they are used today.Why are the Dynkin diagrams E6, E7 and E8 always drawn the way they are drawn?
/ref> Dynkin then used them in 1946 and 1947, acknowledging Coxeter and Witt in his 1947 paper.


Conventions

Dynkin diagrams have been drawn in a number of ways; the convention followed here is common, with 180° angles on nodes of valence 2, 120° angles on the valence 3 node of D''n'', and 90°/90°/180° angles on the valence 3 node of E''n'', with multiplicity indicated by 1, 2, or 3 parallel edges, and root length indicated by drawing an arrow on the edge for orientation. Beyond simplicity, a further benefit of this convention is that diagram automorphisms are realized by Euclidean isometries of the diagrams. Alternative convention include writing a number by the edge to indicate multiplicity (commonly used in Coxeter diagrams), darkening nodes to indicate root length, or using 120° angles on valence 2 nodes to make the nodes more distinct. There are also conventions about numbering the nodes. The most common modern convention had developed by the 1960s and is illustrated in .


Rank 2 Dynkin diagrams

Dynkin diagrams are equivalent to generalized
Cartan matrices In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas th ...
, as shown in this table of rank 2 Dynkin diagrams with their corresponding ''2''x''2'' Cartan matrices. For rank 2, the Cartan matrix form is: : A = \left begin2&a_\\a_&2\end\right /math> A multi-edged diagram corresponds to the nondiagonal Cartan matrix elements -a21, -a12, with the number of edges drawn equal to max(-a21, -a12), and an arrow pointing towards nonunity elements. A generalized Cartan matrix is a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
A = (a_) such that: # For diagonal entries, a_ = 2. # For non-diagonal entries, a_ \leq 0 . # a_ = 0 if and only if a_ = 0 The Cartan matrix determines whether the group is of finite type (if it is a
Positive-definite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
, i.e. all eigenvalues are positive), of affine type (if it is not positive-definite but positive-semidefinite, i.e. all eigenvalues are non-negative), or of indefinite type. The indefinite type often is further subdivided, for example a Coxeter group is Lorentzian if it has one negative eigenvalue and all other eigenvalues are positive. Moreover, multiple sources refer to hyberbolic Coxeter groups, but there are several non-equivalent definitions for this term. In the discussion below, hyperbolic Coxeter groups are a special case of Lorentzian, satisfying an extra condition. For rank 2, all negative determinant Cartan matrices correspond to hyperbolic Coxeter group. But in general, most negative determinant matrices are neither hyperbolic nor Lorentzian. Finite branches have (-a21, -a12)=(1,1), (2,1), (3,1), and affine branches (with a zero determinant) have (-a21, -a12) =(2,2) or (4,1).


Finite Dynkin diagrams


Affine Dynkin diagrams

There are extensions of Dynkin diagrams, namely the affine Dynkin diagrams; these classify Cartan matrices of
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
s. These are classified in , specifically listed on . Affine diagrams are denoted as X_l^, X_l^, or X_l^, where ''X'' is the letter of the corresponding finite diagram, and the exponent depends on which series of affine diagrams they are in. The first of these, X_l^, are most common, and are called extended Dynkin diagrams and denoted with a
tilde The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) i ...
, and also sometimes marked with a + superscript. as in \tilde A_5 = A_5^ = A_5^. The (2) and (3) series are called twisted affine diagrams. Se
Dynkin diagram generator
for diagrams. Here are all of the Dynkin graphs for affine groups up to 10 nodes. Extended Dynkin graphs are given as the ''~'' families, the same as the finite graphs above, with one node added. Other directed-graph variations are given with a superscript value (2) or (3), representing foldings of higher order groups. These are categorized as ''Twisted affine'' diagrams. {, class="wikitable" , + Connected affine Dynkin graphs up to (2 to 10 nodes)
(Grouped as undirected graphs) !Rank !{\tilde{A_{1+} !{\tilde{B_{3+} !{\tilde{C_{2+} !{\tilde{D_{4+} ! E / F / G , - align=center valign=top !2 , {\tilde{A_{1} or {A}_{1}^{(1)}
, rowspan=2,   , {A}_{2}^{(2)}: , rowspan=3,   ,   , - align=center valign=top !3 , {\tilde{A_{2} or {A}_{2}^{(1)}
, {\tilde{C_{2} or {C}_{2}^{(1)}
br>{D}_{5}^{(2)}:
{A}_{4}^{(2)}: , {\tilde{G_{2} or {G}_{2}^{(1)}
br>{D}_{4}^{(3)}

, - align=center valign=top !4 , {\tilde{A_{3} or {A}_{3}^{(1)}
, {\tilde{B_{3} or {B}_{3}^{(1)}
br>{A}_{5}^{(2)}: , {\tilde{C_{3} or {C}_{3}^{(1)}
br>{D}_{6}^{(2)}:
{A}_{6}^{(2)}: ,   , - align=center valign=top !5 , {\tilde{A_{4} or {A}_{4}^{(1)}
, {\tilde{B_{4} or {B}_{4}^{(1)}
br>{A}_{7}^{(2)}: , {\tilde{C_{4} or {C}_{4}^{(1)}
br>{D}_{7}^{(2)}:
{A}_{8}^{(2)}: , {\tilde{D_{4} or {D}_{4}^{(1)}
, {\tilde{F_{4} or {F}_{4}^{(1)}
br>{E}_{6}^{(2)}

, - align=center valign=top !6 , {\tilde{A_{5} or {A}_{5}^{(1)}
, {\tilde{B_{5} or {B}_{5}^{(1)}
br>{A}_{9}^{(2)}: , {\tilde{C_{5} or {C}_{5}^{(1)}
br>{D}_{8}^{(2)}:
{A}_{10}^{(2)}: , {\tilde{D_{5} or {D}_{5}^{(1)}
,   , - align=center valign=top !7 , {\tilde{A_{6} or {A}_{6}^{(1)}
, {\tilde{B_{6} or {B}_{6}^{(1)}

{A}_{11}^{(2)}: , {\tilde{C_{6} or {C}_{6}^{(1)}

{D}_{9}^{(2)}:
{A}_{12}^{(2)}: , {\tilde{D_{6} or {D}_{6}^{(1)}
, {\tilde{E_{6} or {E}_{6}^{(1)}
, - align=center valign=top !8 , {\tilde{A_{7} or {A}_{7}^{(1)}
, {\tilde{B_{7} or {B}_{7}^{(1)}
br>{A}_{13}^{(2)}: , {\tilde{C_{7} or {C}_{7}^{(1)}

{D}_{10}^{(2)}:
{A}_{14}^{(2)}: , {\tilde{D_{7} or {D}_{7}^{(1)}
, {\tilde{E_{7} or {E}_{7}^{(1)}
, - align=center valign=top !9 , {\tilde{A_{8} or {A}_{8}^{(1)}
, {\tilde{B_{8} or {B}_{8}^{(1)}

{A}_{15}^{(2)}: , {\tilde{C_{8} or {C}_{8}^{(1)}

{D}_{11}^{(2)}:
{A}_{16}^{(2)}: , {\tilde{D_{8} or {D}_{8}^{(1)}
, {\tilde{E_{8} or {E}_{8}^{(1)}
, - align=center valign=top !10 , {\tilde{A_{9} or {A}_{9}^{(1)}
, {\tilde{B_{9} or {B}_{9}^{(1)}

{A}_{17}^{(2)}: , {\tilde{C_{9} or {C}_{9}^{(1)}

{D}_{12}^{(2)}:
{A}_{18}^{(2)}: , {\tilde{D_{9} or {D}_{9}^{(1)}
, - align=center valign=top !11 , ... , ... , ... , ...


Hyperbolic and higher Dynkin diagrams

The set of compact and noncompact hyperbolic Dynkin graphs has been enumerated. All rank 3 hyperbolic graphs are compact. Compact hyperbolic Dynkin diagrams exist up to rank 5, and noncompact hyperbolic graphs exist up to rank 10. {, class=wikitable , + Summary , - !Rank !Compact !Noncompact !Total , - !3 , , 31, , 93, , 123 , - !4 , , 3, , 50, , 53 , - !5 , , 1, , 21, , 22 , - !6 , , 0, , 22, , 22 , - !7 , , 0, , 4, , 4 , - !8 , , 0, , 5, , 5 , - !9 , , 0, , 5, , 5 , - !10 , , 0, , 4, , 4


Compact hyperbolic Dynkin diagrams

{, class=wikitable , + Compact hyperbolic graphs , - !colspan=2, Rank 3 !Rank 4 !Rank 5 , - valign=top , Linear graphs * (6 4 2): ** H100(3): ** H101(3): ** H105(3): ** H106(3): * (6 6 2): ** H114(3): ** H115(3): ** H116(3): , Cyclic graphs * (4 3 3): H1(3): * (4 4 3): 3 forms... * (4 4 4): 2 forms... * (6 3 3): H3(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): ** H8(4): ** H13(4): * (4 3 4 3): ** H14(4): , * (4 3 3 3 3): ** H7(5):


Noncompact (Over-extended forms)

Some notations used in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, such as
M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witt ...
, use a "+" superscript for extended groups instead of a "~" and this allows higher extensions groups to be defined. # Extended Dynkin diagrams (affine) are given "+" and represent one added node. (Same as "~") # Over-extended Dynkin diagrams (hyperbolic) are given "^" or "++" and represent two added nodes. # Very-extended Dynkin diagrams with 3 nodes added are given "+++". {, class=wikitable , +Some example over-extended (hyperbolic) Dynkin diagrams , - align=center !Rank !height=30, {AE}_{n} = An-2(1)^ !height=30, {BE}_{n} = Bn-2(1)^
{CE}_{n} !height=30, Cn-2(1)^ !height=30, {DE}_{n} = Dn-2(1)^ ! E / F / G , - align=center !3 , {AE}_{3}: ,   ,   ,   ,   , - align=center !4 , {AE}_{4}:



,   , C2(1)^

A4(2)'^

A4(2)^

D3(2)^
, rowspan=2,   , G2(1)^

D4(3)^
, - align=center !5 , {AE}_{5}:

, {BE}_{5}

{CE}_{5}
, C3(1)^

A6(2)^

A6(2)'^

D5(2)^
, - align=center !6 , {AE}_{6}
, {BE}_{6}

{CE}_{6}
, C4(1)^

A8(2)^

A8(2)'^

D7(2)^
, {DE}_{6}
, F4(1)^

E6(2)^
, - align=center !7 , {AE}_{7}
, {BE}_{7}

{CE}_{7}
, , {DE}_{7}
, - align=center !8 , {AE}_{8}
, {BE}_{8}

{CE}_{8}
, , {DE}_{8}
, E6(1)^
, - align=center !9 , {AE}_{9}
, {BE}_{9}

{CE}_{9}
, , {DE}_{9}
, E7(1)^
, - align=center !10 ,   , {BE}_{10}

{CE}_{10}
, , {DE}_{10}
, {E}_{10}=E8(1)^


238 Hyperbolic groups (compact and noncompact)

The 238 hyperbolic groups (compact and noncompact) of rank n \ge 3 are named as H_i^{(n)} and listed as i=1,2,3... for each rank. File:Rank3CompactHyperbolicDynkins1-31bw.svg File:Rank3NonCompactHyperbolicDynkins32-75bw.svg File:Rank3NonCompactHyperbolicDynkins76-123bw.svg File:Rank4HyperbolicDynkins124-176bw.svg File:Rank5HyperbolicDynkins177-198bw.svg File:Rank6HyperbolicDynkins199-205bw.svg File:Rank6HyperbolicDynkins206-212bw.svg File:Rank6HyperbolicDynkins213-220bw.svg File:Rank7HyperbolicDynkins221-224bw.svg File:Rank8HyperbolicDynkins225-229bw.svg File:Rank9HyperbolicDynkins230-234bw.svg File:Rank10HyperbolicDynkins235-238bw.svg


Very-extended

Very-extended groups are
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
s, defined by adding three nodes to the finite groups. The E8, E7, E6, F4, and G2 offer six series ending as very-extended groups. Other extended series not shown can be defined from An, Bn, Cn, and Dn, as different series for each ''n''. The determinant of the associated Cartan matrix determine where the series changes from finite (positive) to affine (zero) to a noncompact hyperbolic group (negative), and ending as a Lorentz group that can be defined with the use of one time-like dimension, and is used in M theory. {, class=wikitable , + Rank 2 extended series , - !Finite !A_2 !C_2 ! G_2 , - align=center !2 , BGCOLOR="#ffffe0", A2 , BGCOLOR="#ffffe0", C2 , BGCOLOR="#ffffe0", G2 , - align=center !3 , BGCOLOR="#ffe0e0", A2+={\tilde{A_{2}

, BGCOLOR="#ffe0e0", C2+={\tilde{C_{2}

, BGCOLOR="#ffe0e0", G2+={\tilde{G_{2}

, - align=center !4 , BGCOLOR="#e0ffe0", A2++

, BGCOLOR="#e0ffe0", C2++

, BGCOLOR="#e0ffe0", G2++

, - align=center !5 , BGCOLOR="#e0e0ff", A2+++

, BGCOLOR="#e0e0ff", C2+++

, BGCOLOR="#e0e0ff", G2+++

, - align=center !Det(Mn) , 3(3-''n'') , 2(3-''n'') , 3-''n'' {, class=wikitable , + Rank 3 and 4 extended series , - !Finite !A_3 !B_3 !C_3 !A_4 !B_4 !C_4 !D_4 ! F_4 , - align=center !2 , , A12
, , , , , , A2
, - align=center !3 , BGCOLOR="#ffffe0", A3
, BGCOLOR="#ffffe0", B3
, BGCOLOR="#ffffe0", C3
, , B2A1
, , A13
, , - align=center !4 , BGCOLOR="#ffe0e0", A3+={\tilde{A_3
, BGCOLOR="#ffe0e0", B3+={\tilde{B_{3}
, BGCOLOR="#ffe0e0", C3+={\tilde{C_{3}
, BGCOLOR="#ffffe0", A4
, BGCOLOR="#ffffe0", B4
, BGCOLOR="#ffffe0", C4
, BGCOLOR="#ffffe0", D4
, BGCOLOR="#ffffe0", F4
, - align=center !5 , BGCOLOR="#e0ffe0", A3++
, BGCOLOR="#e0ffe0", B3++
, BGCOLOR="#e0ffe0", C3++
, BGCOLOR="#ffe0e0", A4+={\tilde{A_{4}
, BGCOLOR="#ffe0e0", B4+={\tilde{B_{4}
, BGCOLOR="#ffe0e0", C4+={\tilde{C_{4}
, BGCOLOR="#ffe0e0", D4+={\tilde{D_{4}
, BGCOLOR="#ffe0e0", F4+={\tilde{F_{4}
, - align=center !6 , BGCOLOR="#e0e0ff", A3+++
, BGCOLOR="#e0e0ff", B3+++
, BGCOLOR="#e0e0ff", C3+++
, BGCOLOR="#e0ffe0", A4++
, BGCOLOR="#e0ffe0", B4++
, BGCOLOR="#e0ffe0", C4++
, BGCOLOR="#e0ffe0", D4++
, BGCOLOR="#e0ffe0", F4++
, - align=center !7 , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", A4+++
, BGCOLOR="#e0e0ff", B4+++
, BGCOLOR="#e0e0ff", C4+++
, BGCOLOR="#e0e0ff", D4+++
, BGCOLOR="#e0e0ff", F4+++
, - align=center !Det(Mn) , 4(4-''n'') , colspan=2, 2(4-''n'') , 5(5-''n'') , colspan=2, 2(5-''n'') , 4(5-''n'') , 5-''n'' {, class=wikitable , + Rank 5 and 6 extended series , - !Finite !A_5 !B_5 !D_5 !A_6 !B_6 !D_6 !E_6 , - align=center !4 , , B3A1
, A3A1
, , , , A22
, - align=center !5 , BGCOLOR="#ffffe0", A5
, BGCOLOR="#ffffe0", , BGCOLOR="#ffffe0", D5
, , B4A1
, D4A1
, A5
, - align=center !6 , BGCOLOR="#ffe0e0", A5+={\tilde{A_5
, BGCOLOR="#ffe0e0", B5+={\tilde{B_{5}
, BGCOLOR="#ffe0e0", D5+={\tilde{D_5
, BGCOLOR="#ffffe0", A6
, BGCOLOR="#ffffe0", B6
, BGCOLOR="#ffffe0", D6
, BGCOLOR="#ffffe0", E6
, - align=center !7 , BGCOLOR="#e0ffe0", A5++
, BGCOLOR="#e0ffe0", B5++
, BGCOLOR="#e0ffe0", D5++
, BGCOLOR="#ffe0e0", A6+={\tilde{A_6
, BGCOLOR="#ffe0e0", B6+={\tilde{B_{6}
, BGCOLOR="#ffe0e0", D6+={\tilde{D_6
, BGCOLOR="#ffe0e0", E6+={\tilde{E_6
, - align=center !8 , BGCOLOR="#e0e0ff", A5+++
, BGCOLOR="#e0e0ff", B5+++
, BGCOLOR="#e0e0ff", D5+++
, BGCOLOR="#e0ffe0", A6++
, BGCOLOR="#e0ffe0", B6++
, BGCOLOR="#e0ffe0", D6++
, BGCOLOR="#e0ffe0", E6++
, - align=center !9 , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", A6+++
, BGCOLOR="#e0e0ff", B6+++
, BGCOLOR="#e0e0ff", D6+++
, BGCOLOR="#e0e0ff", E6+++
, - align=center !Det(Mn) , 6(6-''n'') , 2(6-''n'') , 4(6-''n'') , 7(7-''n'') , 2(7-''n'') , 4(7-''n'') , 3(7-''n'') {, class=wikitable , + Some rank 7 and higher extended series , - !Finite !A7 !B7 !D7 !E7 ! E8 , - align=center !3 , , , , , E3=A2A1
, - align=center !4 , , , , A3A1
, E4=A4
, - align=center !5 , , , , A5
, E5=D5
, - align=center !6 , , B5A1
, D5A1
, D6
, E6

, - align=center !7 , BGCOLOR="#ffffe0", A7
, BGCOLOR="#ffffe0", B7
, BGCOLOR="#ffffe0", D7
, BGCOLOR="#ffffe0", E7

, E7

, - align=center !8 , BGCOLOR="#ffe0e0", A7+={\tilde{A_7

, BGCOLOR="#ffe0e0", B7+={\tilde{B_{7}

, BGCOLOR="#ffe0e0", D7+={\tilde{D_7

, BGCOLOR="#ffe0e0", E7+={\tilde{E_{7}

, BGCOLOR="#ffffe0", E8
, - align=center !9 , BGCOLOR="#e0ffe0", A7++

, BGCOLOR="#e0ffe0", B7++

, BGCOLOR="#e0ffe0", D7++

, BGCOLOR="#e0ffe0", E7++

, BGCOLOR="#ffe0e0", E9=E8+={\tilde{E_{8}

, - align=center !10 , BGCOLOR="#e0e0ff", A7+++

, BGCOLOR="#e0e0ff", B7+++

, BGCOLOR="#e0e0ff", D7+++

, BGCOLOR="#e0e0ff", E7+++

, BGCOLOR="#e0ffe0", E10=E8++

, - align=center !11 , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", , BGCOLOR="#e0e0ff", E11=E8+++
, - align=center !Det(Mn) , 8(8-''n'') , 2(8-''n'') , 4(8-''n'') , 2(8-''n'') , 9-''n''


See also

*
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
*
List of irreducible Tits indices In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field ''k'', not assumed to be algebraically closed. The possible irreducible indices were ...
* Klassifikation von Wurzelsystemen (Classification of root systems)


Notes


Citations


References

* * * * * * * * * *


External links


John Baez on the ubiquity of Dynkin diagrams in mathematics


{{DEFAULTSORT:Dynkin Diagram Lie algebras