HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 In mathematics, a root system is a configuration of vector space, 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 ...
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.


Lie algebras

In terms of complex semisimple Lie algebras: * A_n corresponds to \mathfrak_(\mathbf), the special linear Lie algebra of traceless operators, * D_n corresponds to \mathfrak_(\mathbf), 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 \mathfrak_, the algebra of the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
SU(n+1); * D_n corresponds to \mathfrak_(\mathbf), the algebra of the even projective special orthogonal group PSO(2n), 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 SU(2), the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced ''affine'' Dynkin diagrams \tilde A_n, \tilde D_n, \tilde E_k, and the representations of these groups can be understood in terms of these diagrams. This connection is known as the after John McKay. The connection to
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s is described in . 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 In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
,
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
/
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
, and
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
/
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
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 group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s A_3, BC_3, and H_3. The
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...
of \mathbf^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 .


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 operator In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a Graph (discrete mathematics), graph or a lattice (group), discrete grid. For the case of a finite-dimensional graph ...
s or Cartan matrices. Proofs in terms of Cartan matrices may be found in . 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: :\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 E8 they range from 58 to 270, and have been observed as early as .


Other classifications

The elementary catastrophes are also classified by the ADE classification. The ADE diagrams are exactly the
quiver A quiver is a container for holding arrows or Crossbow bolt, bolts. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were traditionally made of leath ...
s of finite type, via Gabriel's theorem. There is also a link with
generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles yet containing many quadrangles. A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 a ...
s, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems ''E''6, ''E''7 and ''E''8. 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. It was suggested that symmetries of small
droplet cluster Droplet cluster is a self-assembled levitating monolayer of microdroplets usually arranged into a hexagonally ordered structure over a locally heated thin (about 1 mm) layer of water. The droplet cluster is typologically similar to coll ...
s may be subject to an ADE classification. The minimal models of
two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...
have an ADE classification. Four dimensional \mathcal=2 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 In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the surfa ...
. 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 In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
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 In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
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 mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
in the scope of
Diffeomorphisms In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Defini ...
becomes analogous (and at the same time includes as a special case) the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
of
Symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the ...
.


Trinities

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 Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, which he had previously proposed in the 1970s. In addition to examples from
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
(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 \tilde E_6, \tilde E_7, \tilde 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 \tilde G_2, \tilde F_4, \tilde 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 \tilde E_8 diagram and certain conjugacy classes of the
monster 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; it has order :    : = 2463205976112133171923293 ...
, which is known as ''McKay's E8 observation;'' see also
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
. McKay further relates the nodes of \tilde E_7 to conjugacy classes in 2.''B'' (an order 2 extension of the baby monster group), and the nodes of \tilde E_6 to conjugacy classes in 3.''Fi''24' (an order 3 extension of the
Fischer group In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...
) – 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 group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
s 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 Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...
in the 1830s. In fact, the groups decompose as products of sets (not as products of groups) as: A_4 \times Z_5, S_4 \times Z_7, and A_5 \times Z_. These groups also are related to various geometries, which dates to
Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
in the 1870s; see icosahedral symmetry: related geometries for historical discussion and for more recent exposition. Associated geometries (tilings on
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s) 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 (genus 70) with embedded Paley biplane as an 11-element set (order 3
biplane A biplane is a fixed-wing aircraft with two main wings stacked one above the other. The first powered, controlled aeroplane to fly, the Wright Flyer, used a biplane wing arrangement, as did many aircraft in the early years of aviation. While ...
). 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 Yang-Hui He and John McKay, https://arxiv.org/abs/1505.06742 of relating E8,7,6 with the largest three of the sporadic simple groups, Monster, Baby and Fischer 24', cf.
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
.


See also

* Elliptic surface


References


Sources

* * Problem VIII. The ''A-D-E'' classifications (V. Arnold). * * * * * * * * * *


External links

*
John Baez John Carlos Baez ( ; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, ap ...

This Week's Finds in Mathematical Physics
August 28, 1995, through October 3, 1995, an

May 4, 2006

Tony Smith

Luboš Motl,
The Reference Frame
'' May 7, 2006 {{DEFAULTSORT:Ade Classification Lie groups