ADE classification
Encyclopedia
In mathematics
, the ADE classification (originally A-D-E classifications) is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "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). The list comprises
These comprise 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 isomorphism
s
and corresponding isomorphisms of classified objects.
The question of giving a common origin to these classifications, rather than an a posteriori verification of a parallelism, was posed in .
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.
In terms of compact Lie algebra
s and corresponding simply laced Lie groups:
, 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
s is described in .
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
while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter group
s
and 
The orbifold
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 – see .
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:
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:
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 E8 they range from 58 to 270, and have been observed as early as .
The ADE diagrams are exactly the tame quivers, via Gabriel's theorem
.
There are deep connections between these objects, hinted at by the classification; some of these connections can be understood via string theory
.
" 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 class
es), 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, and those of that ilk, 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
) – note that these are the three largest sporadic group
s, 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
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 surface
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). 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 representation
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the ADE classification (originally A-D-E classifications) is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system
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...
forming angles of
(no edge between the vertices) or (single edge between the vertices). The list comprisesThese comprise 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 isomorphismExceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families of mathematical objects, that is not an example of a pattern of such isomorphisms.Because these series of objects are presented differently, they are not...
s
and corresponding isomorphisms of classified objects.
The question of giving a common origin to these classifications, rather than an a posteriori verification of a parallelism, was posed in .
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 algebraSpecial linear Lie algebraIn mathematics, the special linear Lie algebra of order n is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket [X,Y]:=XY-YX...
of traceless operators, -
corresponds to 
the even special orthogonal Lie algebra of even dimensional skew-symmetric operators, and -
are three of the five exceptional Lie algebras.
In terms of compact Lie algebra
Compact Lie algebra
In the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
s and corresponding simply laced Lie groups:
-
corresponds to 
the algebra of the special unitary groupSpecial unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
-
corresponds to 
the algebra of the even projective special orthogonal group 
, while -
are three of five exceptional compact Lie algebraCompact Lie algebraIn the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
s.
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 McKayJohn McKay (mathematician)
John McKay is a dual British/Canadian citizen, a mathematician at Concordia University, known for his discovery of monstrous moonshine, his joint construction of some sporadic simple groups, for the McKay conjecture in representation theory, and for the McKay correspondence relating certain...
. The connection to Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s is described in .
Note that the ADE correspondence is not the correspondence of Platonic solids to their reflection group
Reflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a...
of symmetries: for instance, in the ADE correspondence the tetrahedron, cube/octahedron, and dodecahedron/icosahedron correspond to
while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter groupCoxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
s
and The orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
of
constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a du Val singularityDu Val singularity
In algebraic geometry, a du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is a singularity of a surface that is a double cover branched over a curve with an A-D-E singularity. They are the canonical singularities in dimension 2...
.
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 – 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 operatorDiscrete 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...
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 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 tame quivers, via Gabriel's theorem
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 representations. classified all quivers of finite type, and also...
.
There are deep connections between these objects, hinted at by the classification; some of these connections can be understood via string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
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 "trinitiesTrinity
The Christian doctrine of the Trinity defines God as three divine persons : the Father, the Son , and the Holy Spirit. The three persons are distinct yet coexist in unity, and are co-equal, co-eternal and consubstantial . Put another way, the three persons of the Trinity are of one being...
" 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 class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es), 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, and those of that ilk, 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 groupMonster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
, which is known as McKay's E8 observation; see also monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
. McKay further relates the nodes of
to conjugacy classes in 2.B (an order 2 extension of the baby monster groupBaby Monster group
In the mathematical field of group theory, the Baby Monster group B is a group of orderThe Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the Monster group...
), and the nodes of
to conjugacy classes in 3.Fi24' (an order 3 extension of the Fischer groupFischer group
In mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
) – note that these are the three largest sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s, 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 GaloisÉvariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
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 KleinFelix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
in the 1870s; see icosahedral symmetry: related geometries for historical discussion and for more recent exposition. Associated geometries (tilings on Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
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
Compound of five tetrahedra
This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.-As a compound:It can be constructed by arranging five tetrahedra in rotational icosahedral symmetry , as colored in the upper right model...
as a 5-element set, PSL(2,7) of the Klein quartic
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
(genus 3) with an embedded (complementary) Fano plane
Fano plane
In finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.-Homogeneous coordinates:...
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). 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 representation
Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
s 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.
External links
- John BaezJohn C. BaezJohn Carlos Baez is an American mathematical physicist and a professor of mathematics at the University of California, Riverside in Riverside, California. He is known for his work on spin foams in loop quantum gravity. More recently, his research has focused on applications of higher categories...
, This Week's Finds in Mathematical Physics: Week 62, Week 63, Week 64, Week 65, August 28, 1995 through October 3, 1995, and Week 230, May 4, 2006 - The McKay Correspondence, Tony Smith
- ADE classification, McKay correspondence, and string theory, Luboš Motl, The Reference Frame, May 7, 2006