
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
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 representat ...
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:
*
corresponds to
the
special linear Lie algebra of
traceless operators,
*
corresponds to
the even
special orthogonal Lie algebra
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
of even-dimensional
skew-symmetric operators, and
*
are three of the five exceptional Lie algebras.
In terms of
compact Lie algebras and corresponding
simply laced Lie groups:
*
corresponds to
the algebra of the
special unitary group
*
corresponds to
the algebra of the even
projective special orthogonal group , while
*
are three of five exceptional
compact Lie algebras.
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 after
John McKay. The connection to
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
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 (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
/
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
, and
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentag ...
/
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...
correspond to
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
and
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 which is locally a finite group quotient of a Euclidean space.
D ...
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 .
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 or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices) ...
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:
:
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 E
8 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, bolts, ammo, projectiles, darts, or javelins. 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 tr ...
s of finite type, via
Gabriel's theorem
In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams.
Statement
A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable represent ...
.
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 ''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.
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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
.
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
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 ...
have an ADE classification.
Four dimensional
superconformal gauge quiver theories with unitary gauge groups have an ADE classification.
Trinities
Arnold
Arnold may refer to:
People
* Arnold (given name), a masculine given name
* Arnold (surname), a German and English surname
Places Australia
* Arnold, Victoria, a small town in the Australian state of Victoria
Canada
* Arnold, Nova Scotia
Uni ...
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
(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 E
8 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.''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 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
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 radicals, ...
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
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
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 ve ...
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
In combinatorics, combinatorial mathematics, a block design is an incidence structure consisting of a set together with a Family of sets, family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain condition ...
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
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...
, 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
Yang-Hui He (; born 29 September 1975) is a mathematical physicist, who is a Fellow at the London Institute, which is based at the Royal Institution of Great Britain, as well as lecturer and former Fellow at Merton College, Oxford. He holds ho ...
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.
See also
* Elliptic surface In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed ...
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, appl ...
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