ADE classification

Lie algebras

In terms of complex semisimple Lie algebras:

A n corresponds to s l n + 1 ( C ) , the special linear Lie algebra of traceless operators,

D n corresponds to s o 2 n ( C ) , the even special orthogonal Lie algebra of even-dimensional skew-symmetric operators, and

E 6 , E 7 , E 8 are three of the five exceptional Lie algebras.

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

A n corresponds to s u n + 1 , the algebra of the special unitary group S U ( n + 1 ) ;

D n corresponds to s o 2 n ( R ) , the algebra of the even projective special orthogonal group P S O ( 2 n ) , while

E 6 , E 7 , E 8 are three of five exceptional compact Lie algebras.

Binary polyhedral groups

The same classification applies to discrete subgroups of S U ( 2 ) , the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin diagrams A ~ n , D ~ n , E ~ k , 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 E 6 , E 7 , E 8 , while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter groups A 3 , B C 3 , and H 3 .

The orbifold of C 2 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, which can be stated in terms of the discrete Laplace operators 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:

Δ ϕ = ϕ .  

Equivalently, the positive functions in the kernel of Δ − I . 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:

Δ ϕ = ϕ − 2.  

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. 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.

There are deep connections between these objects, hinted at by the classification; some of these connections can be understood via string theory and quantum mechanics.

Trinities

Arnold has subsequently proposed many further connections in this vein, under the rubric of "mathematical trinities", and 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. 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 E ~ 6 , E ~ 7 , E ~ 8 (corresponding to tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups S 3 , S 2 , S 1 , respectively, and the associated foldings are the diagrams G ~ 2 , F ~ 4 , E ~ 8 (note that in less careful writing, the extended (tilde) qualifier is often omitted). More significantly, McKay suggests a correspondence between the nodes of the E ~ 8 diagram and certain conjugacy classes of the monster group, which is known as McKay's E₈ observation; see also monstrous moonshine. McKay further relates the nodes of E ~ 7 to conjugacy classes in 2.B (an order 2 extension of the baby monster group), and the nodes of E ~ 6 to conjugacy classes in 3.Fi₂₄' (an order 3 extension of the Fischer group) – 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 A 4 , S 4 , A 5 have connections with the projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660), which is deemed a "McKay correspondence". 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: A 4 × Z 5 , S 4 × Z 7 , and A 5 × Z 11 . 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). 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. 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 of relating E_8,7,6 with the largest three of the Sporadic simple groups, Monster, Baby and Fischer 24', cf. Monstrous Moonshine.