Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, 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, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, 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.

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.

They have the following correspondence for the Lie algebras associated to classical groups over the complex numbers:

For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide.

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.

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.[citation needed]

Lastly, sometimes associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include:

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 simply laced Dynkin diagrams, those with no multiple edges (A, D, E) classify many further mathematical objects; see discussion at ADE classification.

Dynkin diagrams must satisfy certain constraints; these are essentially those satisfied by finite Coxeter–Dynkin diagrams, together with an additional crystallographic constraint.

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:

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.)
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 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.

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.[4]

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.[5][6][7]

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.[6][8]

Disconnected diagrams, which correspond to semisimple Lie algebras, may have automorphisms from exchanging components of the diagram.

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.[9] Further, every multiply laced diagram (finite or infinite) can be obtained by folding a simply-laced diagram.[10]

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.[10] 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 notion of foldings can also be applied more generally to Coxeter diagrams[12] – 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 for further discussion.

Some additional maps of diagrams have meaningful interpretations, as detailed below. However, not all maps of root systems arise as maps of diagrams.[13]

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.

The maximal subgraphs are as follows; subgraphs related by a diagram automorphism are labeled "conjugate":

Finally, duality of diagrams corresponds to reversing the direction of arrows, if any:[13] Bn and Cn are dual, while F4, and G2 are self-dual, as are the simply-laced ADE diagrams.

The simply laced Dynkin diagrams classify diverse mathematical objects; this is called the ADE classification.

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;[14] 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.[citation needed]

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.[14][15] 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;[15] 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).[15]

Dynkin diagrams are equivalent to generalized Cartan matrices, as shown in this table of rank 2 Dynkin diagrams with their corresponding 2x2 Cartan matrices.

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.

The Cartan matrix determines whether the group is of finite type (if it is a Positive-definite matrix, 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).

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.[18]

The set of compact and noncompact hyperbolic Dynkin graphs has been enumerated.[19] 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.

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.

Very-extended groups are Lorentz groups, 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.[20]