ADE classification

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The simply laced Dynkin diagrams classify diverse mathematical objects. Simply Laced Dynkin Diagrams.svg
The simply laced Dynkin diagrams classify diverse mathematical objects.

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, rather than a posteriori verification of a parallelism, was posed in ( Arnold 1976 ). The complete list of simply laced Dynkin diagrams comprises

Contents

Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of (no edge between the vertices) or (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting and ), and three of the five exceptional Dynkin diagrams (omitting and ).

This list is non-redundant if one takes for If one extends the families to include redundant terms, one obtains the exceptional isomorphisms

and corresponding isomorphisms of classified objects.

The A, D, E nomenclature also yields the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.

Lie algebras

In terms of complex semisimple Lie algebras:

In terms of compact Lie algebras and corresponding simply laced Lie groups:

Binary polyhedral groups

The same classification applies to discrete subgroups of , the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin diagrams and the representations of these groups can be understood in terms of these diagrams. This connection is known as the McKay correspondence after John McKay. The connection to Platonic solids is described in ( Dickson 1959 ). The correspondence uses the construction of McKay graph.

Note that the ADE correspondence is not the correspondence of Platonic solids to their reflection group of symmetries: for instance, in the ADE correspondence the tetrahedron, cube/octahedron, and dodecahedron/icosahedron correspond to while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter groups and

The orbifold of constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a du Val singularity.

The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a pair of binary polyhedral groups. This is known as the Slodowy correspondence, named after Peter Slodowy – see ( Stekolshchik 2008 ).

Labeled graphs

The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties, [1] which can be stated in terms of the discrete Laplace operators [2] or Cartan matrices. Proofs in terms of Cartan matrices may be found in ( Kac 1990 , pp. 47–54).

The affine ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property:

Twice any label is the sum of the labels on adjacent vertices.

That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent vertices minus value of vertex) – the positive solutions to the homogeneous equation:

Equivalently, the positive functions in the kernel of The resulting numbering is unique up to scale, and if normalized such that the smallest number is 1, consists of small integers – 1 through 6, depending on the graph.

The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:

Twice any label minus two is the sum of the labels on adjacent vertices.

In terms of the Laplacian, the positive solutions to the inhomogeneous equation:

The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E8 they range from 58 to 270, and have been observed as early as ( Bourbaki 1968 ).

Other classifications

The elementary catastrophes are also classified by the ADE classification.

The ADE diagrams are exactly the quivers of finite type, via Gabriel's theorem.

There is also a link with generalized quadrangles, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems E6, E7 and E8. [3] The classes A and D correspond degenerate cases where the line set is empty or we have all lines passing through a fixed point, respectively. [4]

It was suggested that symmetries of small droplet clusters may be subject to an ADE classification. [5]

The minimal models of two-dimensional conformal field theory have an ADE classification.

Four dimensional superconformal gauge quiver theories with unitary gauge groups have an ADE classification.

Extension of the classification

Arnold has subsequently proposed many further extensions in this classification scheme, in the idea to revisit and generalize the Coxeter classification and Dynkin classification under the single umbrella of root systems. He tried to introduce informal concepts of Complexification and Symplectization based on analogies between Picard–Lefschetz theory which he interprets as the Complexified version of Morse theory and then extend them to other areas of mathematics. He tries also to identify hierarchies and dictionaries between mathematical objects and theories where for example diffeomorphism corresponds to the A type of the Dynkyn classification, volume preserving diffeomorphism corresponds to B type and Symplectomorphisms corresponds to C type. In the same spirit he revisits analogies between different mathematical objects where for example the Lie bracket in the scope of Diffeomorphisms becomes analogous (and at the same time includes as a special case) the Poisson bracket of Symplectomorphism. [6] [7]

Trinities

Arnold extended this further under the rubric of "mathematical trinities". [8] McKay has extended his correspondence along parallel and sometimes overlapping lines. Arnold terms these "trinities" to evoke religion, and suggest that (currently) these parallels rely more on faith than on rigorous proof, though some parallels are elaborated. Further trinities have been suggested by other authors. [9] [10] [11] Arnold's trinities begin with R/C/H (the real numbers, complex numbers, and quaternions), which he remarks "everyone knows", and proceeds to imagine the other trinities as "complexifications" and "quaternionifications" of classical (real) mathematics, by analogy with finding symplectic analogs of classic Riemannian geometry, which he had previously proposed in the 1970s. In addition to examples from differential topology (such as characteristic classes), Arnold considers the three Platonic symmetries (tetrahedral, octahedral, icosahedral) as corresponding to the reals, complexes, and quaternions, which then connects with McKay's more algebraic correspondences, below.

McKay's correspondences are easier to describe. Firstly, the extended Dynkin diagrams (corresponding to tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups respectively, and the associated foldings are the diagrams (note that in less careful writing, the extended (tilde) qualifier is often omitted). More significantly, McKay suggests a correspondence between the nodes of the diagram and certain conjugacy classes of the monster group, which is known as McKay's E8 observation; [12] [13] see also monstrous moonshine. McKay further relates the nodes of to conjugacy classes in 2.B (an order 2 extension of the baby monster group), and the nodes of to conjugacy classes in 3.Fi24' (an order 3 extension of the Fischer group) [13] – note that these are the three largest sporadic groups, and that the order of the extension corresponds to the symmetries of the diagram.

Turning from large simple groups to small ones, the corresponding Platonic groups have connections with the projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660), [14] [15] which is deemed a "McKay correspondence". [16] These groups are the only (simple) values for p such that PSL(2,p) acts non-trivially on p points, a fact dating back to Évariste Galois in the 1830s. In fact, the groups decompose as products of sets (not as products of groups) as: and These groups also are related to various geometries, which dates to Felix Klein in the 1870s; see icosahedral symmetry: related geometries for historical discussion and ( Kostant 1995 ) for more recent exposition. Associated geometries (tilings on Riemann surfaces) in which the action on p points can be seen are as follows: PSL(2,5) is the symmetries of the icosahedron (genus 0) with the compound of five tetrahedra as a 5-element set, PSL(2,7) of the Klein quartic (genus 3) with an embedded (complementary) Fano plane as a 7-element set (order 2 biplane), and PSL(2,11) the buckminsterfullerene surface (genus 70) with embedded Paley biplane as an 11-element set (order 3 biplane). [17] Of these, the icosahedron dates to antiquity, the Klein quartic to Klein in the 1870s, and the buckyball surface to Pablo Martin and David Singerman in 2008.

Algebro-geometrically, McKay also associates E6, E7, E8 respectively with: the 27 lines on a cubic surface, the 28 bitangents of a plane quartic curve, and the 120 tritangent planes of a canonic sextic curve of genus 4. [18] [19] The first of these is well-known, while the second is connected as follows: projecting the cubic from any point not on a line yields a double cover of the plane, branched along a quartic curve, with the 27 lines mapping to 27 of the 28 bitangents, and the 28th line is the image of the exceptional curve of the blowup. Note that the fundamental representations of E6, E7, E8 have dimensions 27, 56 (28·2), and 248 (120+128), while the number of roots is 27+45 = 72, 56+70 = 126, and 112+128 = 240. This should also fit into the scheme [20] of relating E8,7,6 with the largest three of the sporadic simple groups, Monster, Baby and Fischer 24', cf. monstrous moonshine.

See also

Related Research Articles

<span class="mw-page-title-main">Monster group</span> Sporadic simple group

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order

<span class="mw-page-title-main">Root system</span> Geometric arrangements of points, foundational to Lie theory

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 theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

<span class="mw-page-title-main">Dynkin diagram</span> Pictorial representation of symmetry

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. 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 correspond to important features of the associated Lie algebra.

<span class="mw-page-title-main">Simple Lie group</span> Connected non-abelian Lie group lacking nontrivial connected normal subgroups

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.

<span class="mw-page-title-main">Icosahedral symmetry</span> 3D symmetry group

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron and the rhombic triacontahedron.

<span class="mw-page-title-main">Bitangents of a quartic</span> 28 lines which touch a general quartic plane curve in two places

In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane.

<span class="mw-page-title-main">Exceptional object</span>

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others.

<span class="mw-page-title-main">Coxeter–Dynkin diagram</span> Pictorial representation of symmetry

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

<span class="mw-page-title-main">Uniform 8-polytope</span> Polytope contained by 7-polytope facets

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

<span class="mw-page-title-main">Uniform 7-polytope</span> Polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

<span class="mw-page-title-main">Uniform 9-polytope</span> Type of geometric object

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

<span class="mw-page-title-main">Uniform 6-polytope</span> Uniform 6-dimensional polytope

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

<span class="mw-page-title-main">Gosset–Elte figures</span>

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

<span class="mw-page-title-main">Coxeter notation</span> Classification system for symmetry groups in geometry

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

<span class="mw-page-title-main">McKay graph</span> Construction in graph theory

In mathematics, the McKay graph of a finite-dimensional representation V of a finite group G is a weighted quiver encoding the structure of the representation theory of G. Each node represents an irreducible representation of G. If χ i, χ j are irreducible representations of G, then there is an arrow from χ i to χ j if and only if χ j is a constituent of the tensor product Then the weight nij of the arrow is the number of times this constituent appears in For finite subgroups H of the McKay graph of H is the McKay graph of the defining 2-dimensional representation of H.

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

References

  1. ( Proctor 1993 )
  2. ( Proctor 1993 , p. 940)
  3. Cameron P.J.; Goethals, J.M.; Seidel, J.J; Shult, E. E. Line graphs, root systems and elliptic geometry
  4. Godsil Chris; Gordon Royle. Algebraic Graph Theory, Chapter 12
  5. Fedorets A. A., et al. Symmetry of small clusters of levitating water droplets. Phys. Chem. Chem. Phys., 2020, https://doi.org/10.1039/D0CP01804J
  6. Arnold, Vladimir, 1997, Toronto Lectures, Lecture 2: Symplectization, Complexification and Mathematical Trinities, June 1997 (last updated August, 1998). TeX, PostScript, PDF
  7. Polymathematics: is mathematics a single science or a set of arts? On the server since 10-Mar-99, Abstract, TeX, PostScript, PDF; see table on page 8
  8. http://www.neverendingbooks.org/arnolds-trinities [ bare URL ]
  9. Les trinités remarquables, Frédéric Chapoton (in French)
  10. le Bruyn, Lieven (17 June 2008), Arnold's trinities
  11. le Bruyn, Lieven (20 June 2008), Arnold's trinities version 2.0
  12. Arithmetic groups and the affine E8 Dynkin diagram, by John F. Duncan, in Groups and symmetries: from Neolithic Scots to John McKay
  13. 1 2 le Bruyn, Lieven (22 April 2009), the monster graph and McKay's observation
  14. Kostant, Bertram (1995), "The Graph of the Truncated Icosahedron and the Last Letter of Galois" (PDF), Notices Amer. Math. Soc., 42 (4): 959–968, see: The Embedding of PSl(2, 5) into PSl(2, 11) and Galois’ Letter to Chevalier.
  15. le Bruyn, Lieven (12 June 2008), Galois’ last letter, archived from the original on 2010-08-15
  16. ( Kostant 1995 , p. 964)
  17. Martin, Pablo; Singerman, David (April 17, 2008), From Biplanes to the Klein quartic and the Buckyball (PDF)
  18. Arnold 1997, p. 13
  19. ( McKay & Sebbar 2007 , p. 11)
  20. Yang-Hui He and John McKay, https://arxiv.org/abs/1505.06742

Sources