ADE classification

s 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 . The complete list of simply laced Dynkin diagrams comprises : A_n, \, D_n, \, E_6, \, E_7, \, E_8. Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ (no edge between the vertices) or 2\pi/3 = 120^\circ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B_n and C_n ), and three of the five exceptional Dynkin diagrams (omitting F_4 and G_2 ). This list is non-redundant if one takes n \geq 4 for D_n. If one extends the families to include redundant terms, one obtains the exceptional isomorphisms : D_3 \cong A_3, E_4 \cong A_4, E_5 \cong D_5, 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.

s 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 . The complete list of simply laced Dynkin diagrams comprises A_n, \, D_n, \, E_6, \, E_7, \, E_8. Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ (no edge between the vertices) or 2\pi/3 = 120^\circ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B_n and C_n ), and three of the five exceptional Dynkin diagrams (omitting F_4 and G_2 ). This list is non-redundant if one takes n \geq 4 for D_n. If one extends the families to include redundant terms, one obtains the exceptional isomorphisms D_3 \cong A_3, E_4 \cong A_4, E_5 \cong D_5, 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. \Delta \phi = \phi.\ Equivalently, the positive functions in the kernel of \Delta - 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: \Delta \phi = \phi - 2.\ The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E 8 they range from 58 to 270, and have been observed as early as . Four dimensional \mathcal{N}=2 superconformal gauge quiver theories with unitary gauge groups have an ADE classification. Algebro-geometrically, McKay also associates E 6 , E 7 , E 8 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 E 6 , E 7 , E 8 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. ==Sources== Problem VIII. The A-D-E classifications (V. Arnold).

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