Dynkin Classification Essay

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  • 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). The multiple edges are, within certain constraints, directed.

    The main interest in Dynkin diagrams are as a means to classify semisimple Lie algebras over algebraically closed fields. This gives rise to Weyl groups, i.e. to many (although not all) finite reflection groups. Dynkin diagrams may also arise in other contexts.

    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; the and directed diagrams yield the same undirected diagram, correspondingly named In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named.


    Finite Dynkin diagrams

    Affine (extended) Dynkin diagrams

    Classification of semisimple Lie algebras[edit]

    Further information: Semisimple Lie algebra § Classification

    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.

    Related classifications[edit]

    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.
    • 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) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, 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.

    Example: A2[edit]

    For example, the symbol may refer to:

    Construction from root systems[edit]

    Consider a root system, assumed to be reduced and integral (or "crystallographic"). In many applications, this root system will arise from a semisimple Lie algebra. Let be a set of positive simple roots. We then construct a diagram from as follows.[1] Form a graph with one vertex for each element of . 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.[2]) 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 of root systems. Note also that the angles and length ratios between roots are related.[3] 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 , and three edges for a length ratio of . (There are no edges when the roots are orthogonal, regardless of the length ratio.)

    In the A2 root system, shown at right, the roots labeled and form a base. Since these two roots are at angle of 120 degrees (with a length ratio of 1), the Dykin diagram consists of two vertices connected by a single edge: .

    Constraints[edit]

    This section needs expansion. You can help by adding to it.(December 2009)

    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[edit]

    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:

    Partly directed
    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. (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:

    crystallographicpoint group
    directedDynkin diagrams
    undirectedundirected Dynkin diagramsCoxeter 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.

    Isomorphisms[edit]

    Dynkin diagrams are conventionally numbered so that the list is non-redundant: for for for for and starting at 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 or 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.[4]

    Automorphisms[edit]

    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]

    The diagrams that have non-trivial automorphisms are An (), Dn (), 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.[6]

    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.[6][8]

    Disconnected diagrams, which correspond to semisimple 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 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[edit]

    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.

    Folding[edit]

    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]

    The most symmetric Dynkin diagram is D4, which gives rise to triality.
    In characteristic 2, the arrow on F4 can be ignored, yielding an additional diagram automorphism and corresponding Suzuki–Ree groups.
    Finite Coxeter group foldings.

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