v· mathematical field of Lie theory, a Dynkin diagram is a type of graph with some edges doubled or tripled (drawn as a double or triple line), and with any multiple edges directed, satisfying certain constraints. They are of interest firstly because they classify semisimple Lie algebras over algebraically closed fields, and give rise to Weyl groups, which are many (but not all) of the finite reflection groups. They also arise in other contexts. They are named for Eugene Dynkin; see history, below.
There is ambiguity in the terminology: in some cases Dynkin diagrams are assumed directed, in which case they correspond to root systems and semisimple Lie algebras, while in other cases they are assumed undirected, in which case they correspond to Weyl groups; the Bn and Cn directed diagrams yield the same undirected diagram, correspondingly named BCn. In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named.
The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields. One classifies such Lie algebras via their root system, 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 it generates, the so-called Weyl group, and thus undirected Dynkin diagrams classify Weyl groups.
Dynkin diagrams can be interpreted as classifying many distinct, related objects, and the notation "An, Bn, ..." 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 Bn, for instance.
The unoriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, which is the finite reflection group associated to the root system. Thus Bn may refer to the unoriented diagram (a special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxeter group.
Note that while the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots. Beware also that 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 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.
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.
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) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, B4 corresponds to 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.
The A2 root system.
For example, the symbol A2 may refer to:
The Dynkin diagram with 2 connected nodes, , which may also be interpreted as a Coxeter diagram.
The Weyl group of symmetries of the roots (reflections in the hyperplane orthogonal to the roots), isomorphic to the symmetric groupS3 (of order 6).
The abstract Coxeter group, presented by generators and relations,
Dynkin diagrams must satisfy certain constraints; these are essentially those satisfied by finite Coxeter–Dynkin diagrams, together with an additional crystallographic constraint.
Connection with Coxeter diagrams
Dynkin diagrams are closely related to Coxeter diagrams of finite Coxeter groups, and the terminology is often conflated.[note 1]
Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects:
Dynkin diagrams are partly directed – 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.
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.
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:
undirected Dynkin diagrams
Coxeter diagrams of finite groups
By this is meant that Coxeter diagrams of finite groups correspond to point groups 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 Bn and Cn diagrams map to the same undirected diagram, with the resulting Coxeter diagram and Weyl group thus sometimes denoted BCn.
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 H3, H4 and I2(p) for p = 5 p ≥ 7), and correspondingly not every finite Coxeter group is a Weyl group.
The exceptional isomorphisms of connected Dynkin diagrams.
Dynkin diagrams are conventionally numbered so that the list is non-redundant: for An, for Bn, for Cn, for Dn, and En starting at n = 6. The families can however be defined for lower n, yielding exceptional isomorphisms 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:
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.
The most symmetric Dynkin diagram is D4, which gives rise to triality.
In addition to isomorphism between different diagrams, some diagrams also have self-isomorphisms or "automorphisms". Diagram automorphisms correspond to outer automorphisms of the Lie algebra, meaning that the outer automorphism group Out = Aut/Inn equals the group of diagram automorphisms.
The diagrams that have non-trivial automorphisms are An (n > 1), Dn (n > 1), and E6. In all these cases except for D4, there is a single non-trivial automorphism (Out = C2, the cyclic group of order 2), while for D4, the automorphism group is the symmetric group on three letters (S3, 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 An, the diagram automorphism is reversing the diagram, which is a line. The nodes of the diagram index the fundamental weights, which (for An−1) are for , and the diagram automorphism corresponds to the duality Realized as the Lie algebra the outer automorphism can be expressed as negative transpose, , which is how the dual representation acts.
For Dn, the diagram automorphism is switching the two nodes at the end of the Y, and corresponds to switching the two chiralspin representations. Realized as the Lie algebra the outer automorphism can be expressed as conjugation by a matrix in O(2n) with determinant −1. Note that so their automorphisms agree, while which is disconnected, and the automorphism corresponds to switching the two nodes.
For D4, the fundamental representation is isomorphic to the two spin representations, and the resulting symmetric group on three letter (S3, or alternatively the dihedral group 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 semisimple Lie algebras, may have automorphisms from exchanging components of the diagram.
In characteristic 2, the arrow on F4 can be ignored, yielding an additional diagram automorphism and corresponding Suzuki–Ree groups.
In positive characteristic there are additional diagram automorphisms – roughly speaking, in characteristic p one is 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 and of F4, while in characteristic 3 there is an order 2 automorphism of G2.
Construction of Lie groups via diagram automorphisms
Diagram automorphisms in turn yield additional Lie groups and groups of Lie type, which are of central importance in the classification of finite simple groups.
The Chevalley group 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 2An, while the other orthogonal groups are constructed as 2Dn, 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.
A (simply-laced) Dynkin diagram (finite or affine) 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.
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 automorphism of A2n does not yield a folding because the middle two nodes are connected by an edge, but in the same orbit.)
Dn+1 → Bn
D4 → G2 (if quotienting by the full group or a 3-cycle, in addition to D4 → C3 in 3 different ways, if quotienting by an involution)
E6 → F4
Similar foldings exist for affine diagrams, including:
The notion of foldings can also be applied more generally to Coxeter diagrams – notably, one can generalize allowable quotients of Dynkin diagrams to Hn and I2(p). Geometrically this corresponds to projections of uniform polytopes. Notably, any simply laced Dynkin diagram can be folded to I2(h), where h is the Coxeter number, which corresponds geometrically to projection to the Coxeter plane.
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. See Math Overflow: Folding by Automorphisms for further discussion.
Other maps of diagrams
A2 root system
G2 root system
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 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 Dn and En). At the level of Lie algebras, these inclusions correspond to sub-Lie algebras.
A Dynkin diagram with no multiple edges is called simply laced, as are the corresponding Lie algebra and Lie group. These are the An,Dn,En diagrams, and phenomena that such diagrams classify are referred to as an ADE classification. In this case the Dynkin diagrams exactly coincide with Coxeter diagrams, as there are no multiple edges.
Main article: Satake diagram
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 diagrams, 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.
Dynkin diagrams are named for Eugene Dynkin, who used them in two papers (1946, 1947) simplifying the classification of semisimple Lie algebras; see (Dynkin 2000). 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 groups, 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. Dynkin then used them in 1946 and 1947, acknowledging Coxeter and Witt in his 1947 paper.
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 Dn, and 90°/90°/180° angles on the valence 3 node of En, 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 (Bourbaki 1968).
Rank 2 Dynkin diagrams
Dynkin diagrams are equivalent to generalized Cartan matrices, as shown in this table of rank 2 Dynkin diagrams with their corresponding 2x2 Cartan matrices.
For rank 2, the Cartan matrix form is:
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 A = (aij) such that:
For diagonal entries, aii = 2.
For non-diagonal entries, .
aij = 0 if and only if aji = 0
The determinant of the Cartan matrix determines whether the group is finite, affine, or hyperbolic. (Note: For rank 2, all negative determinant matrices represent hyperbolic groups. For rank 3 or higher, most negative determine matrices are not hyperbolic, and are instead considered 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).
For more details on this topic, see Affine root system.
There are extensions of Dynkin diagrams, namely the affine Dynkin diagrams; these classify Cartan matrices of affine Lie algebras. These are classified in (Kac 1994, Chapter 4, pp. 47–), specifically listed on (Kac 1994, pp. 53–55). Affine diagrams are denoted as or 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, are most common, and are called extended Dynkin diagrams and denoted with a tilde, and also sometimes marked with a + superscript. as in . The (2) and (3) series are called twisted affine diagrams.
The set of extended affine Dynkin diagrams, with added nodes in green
"Twisted" affine forms are named with (2) or (3) superscripts.
(k is the number of nodes in the graph)
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.
Connected affine Dynkin graphs up to (2 to 10 nodes)
(Grouped as undirected graphs)
E / F / G
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.
The 238 enumerated hyperbolic groups (compact and noncompact) are named as: Hi(n), for rank n, and counting i=1,2,3... for each rank.
Compact hyperbolic Dynkin diagrams
Compact hyperbolic graphs
(6 4 2):
(6 6 2):
(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):
(4 3 4 3):
(4 3 3 3 3):
Noncompact (Over-extended forms)
Some notations used in theoretical physics, such as M-theory, 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 "+++".
Some example over-extended (hyperbolic) Dynkin diagrams
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