Editing ADE classification

[[File:Simply Laced Dynkin Diagrams.svg|thumb|The [[simply laced Dynkin diagram]]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 diagram]]s. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in {{Harv|Arnold|1976}}. The complete list of [[simply laced Dynkin diagram]]s comprises
:<math>A_n, \, D_n, \, E_6, \, E_7, \, E_8.</math>

Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the [[root system]] forming angles of <math>\pi/2 = 90^\circ</math> (no edge between the vertices) or <math>2\pi/3 = 120^\circ</math> (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting <math>B_n</math> and <math>C_n</math>), and three of the five exceptional Dynkin diagrams (omitting <math>F_4</math> and <math>G_2</math>).

This list is non-redundant if one takes <math>n \geq 4</math> for <math>D_n.</math> If one extends the families to include redundant terms, one obtains the [[exceptional isomorphism]]s
:<math>D_3 \cong A_3, E_4 \cong A_4, E_5 \cong D_5,</math>
and corresponding isomorphisms of classified objects.

The ''A'', ''D'', ''E'' nomenclature also yields the simply laced [[finite Coxeter group]]s, 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:
* <math>A_n</math> corresponds to <math>\mathfrak{sl}_{n+1}(\mathbf{C}),</math> the [[special linear Lie algebra]] of [[traceless]] operators, 
* <math>D_n</math> corresponds to <math>\mathfrak{so}_{2n}(\mathbf{C}),</math> the even [[special orthogonal Lie algebra]] of even-dimensional [[skew-symmetric matrix|skew-symmetric]] operators, and
* <math>E_6, E_7, E_8</math> are three of the five exceptional Lie algebras.

In terms of [[compact Lie algebra]]s and corresponding [[simple Lie group#Simply laced groups|simply laced Lie group]]s:
* <math>A_n</math> corresponds to <math>\mathfrak{su}_{n+1},</math> the algebra of the [[special unitary group]] <math>SU(n+1);</math>
* <math>D_n</math> corresponds to <math>\mathfrak{so}_{2n}(\mathbf{R}),</math> the algebra of the even [[projective special orthogonal group]] <math>PSO(2n)</math>, while
* <math>E_6, E_7, E_8</math> are three of five exceptional [[compact Lie algebra]]s.

== Binary polyhedral groups ==
The same classification applies to discrete subgroups of <math>SU(2)</math>, the [[binary polyhedral group]]s; properly, binary polyhedral groups correspond to the simply laced ''affine'' [[Dynkin diagram]]s <math>\tilde A_n, \tilde D_n, \tilde E_k,</math> and the representations of these groups can be understood in terms of these diagrams. This connection is known as the '''{{visible anchor|McKay correspondence}}''' after [[John McKay (mathematician)|John McKay]]. The connection to [[Platonic solid]]s is described in {{Harv|Dickson|1959}}. 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]], [[cube]]/[[octahedron]], and [[dodecahedron]]/[[icosahedron]] correspond to <math>E_6, E_7, E_8,</math> while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the [[Coxeter group]]s <math>A_3, BC_3,</math> and <math>H_3.</math>

The [[orbifold]] of <math>\mathbf{C}^2</math> 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 {{Harv|Stekolshchik|2008}}.

== Labeled graphs ==
The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties,<ref>{{Harv|Proctor|1993}}</ref> which can be stated in terms of the [[discrete Laplace operator]]s<ref>{{Harv|Proctor|1993|loc = p. 940}}</ref> or [[Cartan matrices]]. Proofs in terms of Cartan matrices may be found in {{Harv|Kac|1990|loc = pp. 47–54}}.

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:
:<math>\Delta \phi = \phi.\ </math>
Equivalently, the positive functions in the kernel of <math>\Delta - I.</math> 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:
:<math>\Delta \phi = \phi - 2.\ </math>
The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E<sub>8</sub> they range from 58 to 270, and have been observed as early as {{Harv|Bourbaki|1968}}.

== Other classifications ==
The [[Catastrophe theory#Elementary catastrophes|elementary catastrophes]] are also classified by the ADE classification.

The ADE diagrams are exactly the [[quiver (mathematics)|quiver]]s of finite type, via [[Gabriel's theorem]].

There is also a link with [[generalized quadrangle]]s, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems ''E''<sub>6</sub>, ''E''<sub>7</sub> and ''E''<sub>8</sub>.<ref>Cameron P.J.; Goethals, J.M.; Seidel, J.J; Shult, E. E. ''Line graphs, root systems and elliptic geometry''</ref>
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.<ref>Godsil Chris; Gordon Royle. ''Algebraic Graph Theory'', Chapter 12</ref>

It was suggested that symmetries of small [[droplet cluster]]s may be subject to an ADE classification.<ref>Fedorets A. A., et al. Symmetry of small clusters of levitating water droplets. ''Phys. Chem. Chem. Phys.'', 2020, https://doi.org/10.1039/D0CP01804J</ref>

The [[Minimal model (physics)|minimal models]] of [[two-dimensional conformal field theory]] have an ADE classification.

Four dimensional <math>\mathcal{N}=2</math> superconformal gauge quiver theories with unitary gauge groups have an ADE classification.

== Extension of the classification ==
[[Vladimir Arnold|Arnold]] has subsequently proposed many further extensions in this classification scheme, in the idea to revisit and generalize the [[Coxeter-Dynkin diagram|Coxeter classification]] and [[Dynkin diagram|Dynkin classification]] under the single umbrella of [[root systems]].
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]] 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]] corresponds to the A type of the [[Dynkin diagrams|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 the scope of [[Diffeomorphisms]] becomes analogous (and at the same time includes as a special case) the [[Poisson bracket]] of [[Symplectomorphism]].<ref>Arnold, Vladimir, 1997, Toronto Lectures, ''[http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html Lecture 2: Symplectization, Complexification and Mathematical Trinities],'' June 1997 (last updated August, 1998). [http://www.pdmi.ras.ru/~arnsem/Arnold/a2src.zip TeX], [http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect2.ps.gz PostScript], [http://www.maths.ed.ac.uk/~aar/papers/arnold4.pdf PDF]</ref><ref>''[http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html Polymathematics: is mathematics a single science or a set of arts?]'' On the server since 10-Mar-99, [http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.txt Abstract], [http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.tex.gz TeX], [http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.ps.gz PostScript], [http://www.neverendingbooks.org/DATA/ArnoldPolymathics.pdf PDF]; see table on page 8</ref> 

== Trinities ==
Arnold extended this further under the rubric of "mathematical trinities".<ref name="arntrin"/> McKay has extended his correspondence along parallel and sometimes overlapping lines. Arnold terms these "[[Trinity|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.<ref>''[http://math.univ-lyon1.fr/~chapoton/trinites.html Les trinités remarquables],'' [http://math.univ-lyon1.fr/~chapoton/ Frédéric Chapoton] {{in lang|fr}}</ref><ref name="arntrin">{{citation | last = le Bruyn | first = Lieven | title = Arnold's trinities | url = http://www.neverendingbooks.org/arnolds-trinities | date = 17 June 2008 }}</ref><ref>{{citation | last = le Bruyn | first = Lieven | title = Arnold's trinities version 2.0 | url = http://www.neverendingbooks.org/arnolds-trinities-version-20 | date = 20 June 2008 }}</ref> 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 correspondence|McKay's correspondences]] are easier to describe. Firstly, the extended Dynkin diagrams <math>\tilde E_6, \tilde E_7, \tilde E_8</math> (corresponding to tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups <math>S_3, S_2, S_1,</math> respectively, and the associated [[folding (Dynkin diagram)|foldings]] are the diagrams <math>\tilde G_2, \tilde F_4, \tilde E_8</math> (note that in less careful writing, the extended (tilde) qualifier is often omitted). More significantly, McKay suggests a correspondence between the nodes of the <math>\tilde E_8</math> diagram and certain conjugacy classes of the [[monster group]], which is known as ''McKay's E<sub>8</sub> observation;''<ref>[http://arxiv4.library.cornell.edu/abs/0810.1465 Arithmetic groups and the affine E<sub>8</sub> Dynkin diagram], by John F. Duncan, in ''Groups and symmetries: from Neolithic Scots to John McKay''</ref><ref name="monster">{{citation | last = le Bruyn | first = Lieven | title = the monster graph and McKay's observation | url = http://www.neverendingbooks.org/the-monster-graph-and-mckays-observation | date = 22 April 2009 }}</ref> see also [[monstrous moonshine]]. McKay further relates the nodes of <math>\tilde E_7</math> to conjugacy classes in 2.''B'' (an order 2 extension of the [[baby monster group]]), and the nodes of <math>\tilde E_6</math> to conjugacy classes in 3.''Fi''<sub>24</sub>' (an order 3 extension of the [[Fischer group]])<ref name="monster" /> – 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 <math>A_4, S_4, A_5</math> have connections with the [[projective special linear group]]s PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660),<ref>{{Citation | id = see: The Embedding of PSl(2, 5) into PSl(2, 11) and Galois’ Letter to Chevalier. | title = The Graph of the Truncated Icosahedron and the Last Letter of Galois | first = Bertram | last = Kostant | journal = Notices Amer. Math. Soc. | volume = 42 | number = 4 | pages = 959–968 | year = 1995 | url = https://www.ams.org/notices/199509/kostant.pdf }}</ref><ref>{{citation|last=le Bruyn |first=Lieven |url=http://www.neverendingbooks.org/galois-last-letter |title=Galois' last letter |date=12 June 2008 |url-status=live |archive-url=https://web.archive.org/web/20100815034546/http://www.neverendingbooks.org/index.php/galois-last-letter.html |archive-date=2010-08-15 }}</ref> which is deemed a "McKay correspondence".<ref>{{Harv|Kostant|1995|loc=p. 964}}</ref> These groups are the only (simple) values for ''p'' such that PSL(2,''p'') [[Projective linear group#Action on p points|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: <math>A_4 \times Z_5,</math> <math>S_4 \times Z_7,</math> and <math>A_5 \times Z_{11}.</math> These groups also are related to various geometries, which dates to [[Felix Klein]] in the 1870s; see [[Icosahedral symmetry#Related geometries|icosahedral symmetry: related geometries]] for historical discussion and {{Harv|Kostant|1995}} 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 '''{{visible anchor|buckminsterfullerene surface}}''' (genus 70) with embedded [[Paley biplane]] as an 11-element set (order 3 [[biplane geometry|biplane]]).<ref name="martin">{{ citation | last1 = Martin | first1 = Pablo | last2 = Singerman | first2 = David | title = From Biplanes to the Klein quartic and the Buckyball | url = http://www.neverendingbooks.org/DATA/biplanesingerman.pdf | date = April 17, 2008 }}</ref> 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 E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub> respectively with: the [[27 lines on a cubic surface]], the 28 [[Bitangents of a quartic|bitangents of a plane quartic curve]], and the 120 tritangent planes of a canonic sextic curve of genus 4.<ref>Arnold 1997, p. 13</ref><ref>{{Harv|McKay|Sebbar|2007|p=11}}</ref> 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 E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub> 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 <ref>[[Yang-Hui He]] and [[John McKay (mathematician)|John McKay]], https://arxiv.org/abs/1505.06742</ref> of relating E<sub>8,7,6</sub> with the largest three of the sporadic simple groups, Monster, Baby and Fischer 24', cf. [[monstrous moonshine]].

== See also ==

* [[Elliptic surface]]

== References ==
{{Reflist}}

==Sources==
{{refbegin}}
* {{citation
| author-link = Nicolas Bourbaki
| first = Nicolas | last = Bourbaki | year = 1968 | title = Groupes et algebres de Lie | chapter = Chapters 4–6 | publisher = Hermann | location = Paris }}
* {{Citation | author-link = Vladimir Arnold | first = Vladimir | last = Arnold | page = [https://books.google.com/books?id=BLnRsA-wRsoC&pg=PA46 46] | year = 1976 | title = Mathematical developments arising from Hilbert problems | publisher = [[American Mathematical Society]] | editor = Felix E. Browder | editor-link = Felix E. Browder | chapter = Problems in present day mathematics | series = Proceedings of symposia in pure mathematics | volume = 28}} Problem VIII. The ''A-D-E'' classifications (V. Arnold).
* {{Citation | chapter=XIII: Groups of the Regular Solids; Quintic Equations | chapter-url = https://books.google.com/books?id=RmlOoBznB8wC&pg=PA220| author-link = Leonard Eugene Dickson | first = Leonard E. | last = Dickson | title = Algebraic Theories | publisher = Dover Publications | location= New York | year = 1959 | isbn=978-0-486-15520-3 }}
* {{Citation | last1=Hazewinkel | first1=Michiel | author-link = Michiel Hazewinkel | last2 = Hesseling | first3 = JD. | last3 = Siersma | first4 = F. | last4 = Veldkamp | title = The ubiquity of Coxeter Dynkin diagrams. (An introduction of the A-D-E problem) | journal = Nieuw Archief V. Wiskunde | volume = 35 | number = 3 | year = 1977 | pages = 257–307 | url = http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf }}
* {{Citation | first = John | last = McKay  | author-link = John McKay (mathematician) | title = Graphs, singularities and finite groups | journal = Proc. Symp. Pure Math. | series = Proceedings of Symposia in Pure Mathematics | volume = 37 | publisher = Amer. Math. Soc. | year  = 1980 | pages = 183– and 265–| doi = 10.1090/pspum/037/604577 | isbn = 978-0-8218-1440-6 }}
* {{Citation | first = John | last = McKay  | author-link = John McKay (mathematician) | chapter = Representations and Coxeter Graphs |title = "The Geometric Vein", Coxeter Festschrift | year = 1982 | publisher = [[Springer-Verlag]] | location = Berlin | pages= 549–}}
* {{ citation | first = Victor G. | last = Kac | title = Infinite-Dimensional Lie Algebras | edition = 3rd | publisher = [[Cambridge University Press]] | location = Cambridge | year = 1990 | isbn = 0-521-46693-8 }}
* {{Citation | last = McKay | first = John | author-link = John McKay (mathematician) | url = http://math.ucr.edu/home/baez/ADE.html | title = A Rapid Introduction to ADE Theory | date =  January 1, 2001  }}
* {{Citation
| doi = 10.2307/2324217
| issn = 0002-9890
| volume = 100
| issue = 10
| pages = 937–941
| last = Proctor | first = R. A.
| title = Two Amusing Dynkin Diagram Graph Classifications
| journal = [[The American Mathematical Monthly]]| date=December 1993
| jstor = 2324217
}}
* {{Cite book| doi = 10.1007/978-3-540-30308-4_10 | title=Frontiers in Number Theory, Physics, and Geometry, II | year = 2007| contribution=Replicable Functions: An introduction| publisher=Springer| last1 = McKay | first1 = J. | last2 = Sebbar | first2 = Abdellah| pages = 373–386| isbn=978-3-540-30307-7 }}
* {{Citation
| title = Notes on Coxeter Transformations and the McKay Correspondence | series = Springer Monographs in Mathematics | first = R. | last = Stekolshchik
| year = 2008 | isbn = 978-3-540-77398-6
| doi = 10.1007/978-3-540-77398-3 
| doi-broken-date = 15 April 2025 }}
* {{Citation | first = Joris | last = van Hoboken | title = Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type A,D,E | series = Master's Thesis | publisher = University of Amsterdam | year = 2002 | url = http://www.jorisvanhoboken.nl/wp-content/uploads/2007/03/platonic-solids-binary-polyhedral-groups-kleinian-singularities-and-lie-algebras-of-type-ade.pdf | access-date = 2011-11-23 | archive-url = https://web.archive.org/web/20120426001310/http://www.jorisvanhoboken.nl/wp-content/uploads/2007/03/platonic-solids-binary-polyhedral-groups-kleinian-singularities-and-lie-algebras-of-type-ade.pdf | archive-date = 2012-04-26 | url-status = dead }}
{{refend}}

== External links ==
* [[John C. Baez|John Baez]], [http://math.ucr.edu/home/baez/TWF.html This Week's Finds in Mathematical Physics]: [http://math.ucr.edu/home/baez/week62.html Week 62], [http://math.ucr.edu/home/baez/week63.html Week 63], [http://math.ucr.edu/home/baez/week64.html Week 64], [http://math.ucr.edu/home/baez/week65.html Week 65], August 28, 1995, through October 3, 1995, and [http://math.ucr.edu/home/baez/week230.html Week 230], May 4, 2006 
* [http://www.valdostamuseum.com/hamsmith/McKay.html The McKay Correspondence], Tony Smith
* [http://motls.blogspot.com/2006/05/ade-classification-mckay.html ADE classification, McKay correspondence, and string theory], [[Luboš Motl]], ''[http://motls.blogspot.com/ The Reference Frame],'' May 7, 2006

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[[Category:Lie groups]]