ADE классификациясы: математикалық байланыстар мен үлгілер

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

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.

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

Arnold extended this further under the rubric of "mathematical trinities". 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 (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; 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) 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: 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 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 E8,7,6 with the largest three of the sporadic simple groups, Monster, Baby and Fischer 24', cf. monstrous moonshine.