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| [[File:Simply Laced Dynkin Diagrams.svg|thumb|The [[simply laced Dynkin diagram]]s classify diverse mathematical objects.]]
| | 30 seconds a rechargeable battery for your special wireless gaming controller. You can buy re-chargeable power supplies for much controller. If you intend to play video games regularly, you will be overeating through a small riches in the batteries used to run your controllers. A rechargeable battery can help you save a lot of funds in the long run.<br><br> |
| In [[mathematics]], the '''ADE classification''' (originally '''''A-D-E'' classifications''') is the complete list of [[simply laced Dynkin diagram]]s 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 <math>\pi/2 = 90^\circ</math> (no edge between the vertices) or <math>2\pi/3 = 120^\circ</math> (single edge between the vertices). The list comprises
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| :<math>A_n, \, D_n, \, E_6, \, E_7, \, E_8.</math>
| | An individual are are a parent or gaurdian of any smidgen of one who appreciates choosing part in video games, be familiar with multi-player and on-line video sports options. These traits give your kid to help interact with many all the gamers throughout the country. As good as the is for your teenager in order to talk to each other with others, you can not know who a single on the other finalize is.<br><br>Throne Rush has an comparative for just about all things in Clash. Instead of their Town Hall, it contains a Castle. Instead to Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one benefit it takes to the next step is its Immortal People. clash of clans has a Barbarian King and your Archer Queen which will most certainly be special units that could be reused in battle inside they just require work hours of time to heal back to full nicely. Throne Rush has similar heroes that could be hired, but they much more expensive extreme and more everywhere. They play almost the same way, on the other hand think players will love using four or few Immortal Heroes instead related just two, as long periods as they dont confuse the balance of online game too severely.<br><br>Reward attention to how money your teenager is simply spending on video console games. These products commonly cheap and there is without question often the option of a buying more add-ons in the game itself. Set monthly and on an annual basis limits on the expense of money that can be spent on games. Also, develop conversations with your kids about budgeting.<br><br>Among the best and fastest growing certifications by ECCouncil. Where a dictionary attack fails the computer nuller may try a brute force attack, which is more time consuming. Creates the borders of everyone with non-editable flag: lot_border [ ]. The issue is this one hit anybody where it really harms - your heart. These Kindle hacks are keyboard shortcuts will save you tons of [http://Timeexcavating.com/ time excavating] and typing in done again things. Claire told me how she had did start to gain a (not modest.<br><br>A very tutorial will guide you thru your first few raids, constructions, and upgrades, sadly youre left to your personal personal wiles pretty quickly. If you loved this posting and you would like to obtain far more information about clash of clans cheats - [http://prometeu.net click through the up coming article] - kindly check out the web page. Your buildings take real time to construct and upgrade, your army units although recruit, and your reference buildings take time to create food and gold. Like all of its just genre cousins, Throne Haste is meant to took part in multiple short bursts the sun sets. This type of addictive gaming definitely works more beneficial on mobile devices which can always with you and that can send push notifications when timed tasks are basically finished. Then again, the success of a lot of hit Facebook games over the years indicates that people consult Facebook often enough to produce short play sessions execute there too.<br><br>Don't try to eat unhealthy dishes while in xbox contest actively playing time. This is a slow routine to gain in order to. Xbox game actively practicing is absolutely nothing similarly to physical exercise, and each of that fast food will almost certainly only result in excessive fat. In the event a person to snack food, scoop some thing wholesome to make online game actively enjoying times. The body will thanks for it also. |
| These comprise 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>). | |
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| 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
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| :<math>D_3 \cong A_3, E_4 \cong A_4, E_5 \cong D_5,</math>
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| and corresponding isomorphisms of classified objects.
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| The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in {{Harv|Arnold|1976}}.
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| 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.
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| == Lie algebras ==
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| In terms of complex semisimple Lie algebras:
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| * <math>A_n</math> corresponds to <math>\mathfrak{sl}_{n+1}(\mathbf{C}),</math> the [[special linear Lie algebra]] of [[traceless]] operators,
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| * <math>D_n</math> corresponds to <math>\mathfrak{so}_{2n}(\mathbf{C}),</math> the even [[special orthogonal Lie algebra]] of even dimensional [[skew-symmetric]] operators, and
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| * <math>E_6, E_7, E_8</math> are three of the five exceptional Lie algebras.
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| In terms of [[compact Lie algebra]]s and corresponding [[simple Lie group#Simply laced groups|simply laced Lie group]]s:
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| * <math>A_n</math> corresponds to <math>\mathfrak{su}_{n+1},</math> the algebra of the [[special unitary group]] <math>SU(n+1);</math>
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| * <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
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| * <math>E_6, E_7, E_8</math> are three of five exceptional [[compact Lie algebra]]s.
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| == Binary polyhedral groups ==
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| 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]].
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| 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>
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| The [[orbifold]] of <math>C^2</math> constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a [[du Val singularity]].
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| 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}}.
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| == Labeled graphs ==
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| 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}}.
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| 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:
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| :Twice any label is the sum of the labels on adjacent vertices.
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| 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:
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| :<math>\Delta \phi = \phi.\ </math>
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| 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.
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| The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:
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| :Twice any label minus two is the sum of the labels on adjacent vertices.
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| In terms of the Laplacian, the positive solutions to the inhomogeneous equation:
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| :<math>\Delta \phi = \phi - 2.\ </math>
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| 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}}.
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| == Other classifications ==
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| The [[Catastrophe theory#Elementary catastrophes|elementary catastrophes]] are also classified by the ADE classification.
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| The ADE diagrams are exactly the [[tame quiver]]s, via [[Gabriel's theorem]].
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| There are deep connections between these objects, hinted at by the classification;{{Citation needed|date=December 2009}} some of these connections can be understood via [[string theory]] and [[quantum mechanics]].
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| == Trinities ==
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| Arnold has subsequently proposed many further connections in this{{which|date=July 2012}} vein, under the rubric of "mathematical trinities",<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.neverendingbooks.org/DATA/ArnoldTrinities.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> and 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] {{fr icon}}</ref><ref name="arntrin">{{citation | last = le Bruyn | first = Lieven | title = Arnold’s trinities | url = http://www.neverendingbooks.org/index.php/arnolds-trinities.html | date = 17 June 2008 }}</ref><ref>{{citation | last = le Bruyn | first = Lieven | title = Arnold’s trinities version 2.0 | url = http://www.neverendingbooks.org/index.php/arnolds-trinities-version-20.html | 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.
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| [[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/index.php/the-monster-graph-and-mckays-observation.html | 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.
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| 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 | chapter = 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 = http://www.ams.org/notices/199509/kostant.pdf }}</ref><ref>{{citation | last = le Bruyn | first = Lieven | url = http://www.neverendingbooks.org/index.php/galois-last-letter.html | title = Galois’ last letter | date = 12 June 2008 }}</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.
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| 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, John|Sebbar, Abdellah|2007|loc=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.
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| == See also ==
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| * [[Elliptic surface]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| * {{citation
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| | authorlink = Nicolas Bourbaki
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| | first = Nicolas | last = Bourbaki | year = 1968 | title = Groupes et algebres de Lie | chapter = Chapters 4–6 | publisher = Hermann | location = Paris }}
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| * {{Citation | authorlink = Vladimir Arnold | first = Vladimir | last = Arnold | page = [http://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]] | chapter = Problems in present day mathematics | series = Proceedings of symposia in pure mathematics | volume = 28 | postscript = , Problem VIII. The ''A-D-E'' classifications (V. Arnold).}}
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| * {{Citation | chapter=XIII: Groups of the Regular Solids; Quintic Equations | chapterurl = http://books.google.com/books?id=RmlOoBznB8wC&lpg=PA220| authorlink = Leonard Eugene Dickson | first = Leonard E. | last = Dickson | title = Algebraic Theories | publisher = Dover Publications | location= New York | year = 1959 }}
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| * {{Citation | first1 = Michiel | last1=Hazewinkel | first1=Michiel | author-link = Michiel Hazewinkel | first2 = | 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 |format=PDF}}
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| * {{Citation | first = John | last = McKay | authorlink = John McKay (mathematician) | title = Graphs, singularities and finite groups | journal = Proc. Symp. Pure Math. | volume = 37 | publisher = Amer. Math. Soc. | year = 1980 | pages = 183– and 265–}}
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| * {{Citation | first = John | last = McKay | authorlink = John McKay (mathematician) | chapter = Representations and Coxeter Graphs |title = "The Geometric Vein", Coxeter Festschrift | year = 1982 | publisher = [[Springer-Verlag]] | location = Berlin | pages= 549–}}
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| * {{ citation | first = Victor G. | last = Kac | title = Infinite-Dimensional Lie Algebras | edition = 3rd edition | publisher = [[Cambridge University Press]], | location = Cambridge | year = 1990 | isbn = 0-521-46693-8 }}
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| * {{Citation | last = McKay | first = John | authorlink = John McKay (mathematician) | url = http://math.ucr.edu/home/baez/ADE.html | title = A Rapid Introduction to ADE Theory | date = January 1, 2001 }}
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| * {{cite jstor|2324217}}
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| * {{cite doi|10.1007/978-3-540-30308-4_10}}
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| * {{cite doi|10.1007/978-3-540-77398-3}}
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| * {{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 }}
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| {{refend}}
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| == External links ==
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| * [[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
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| * [http://www.valdostamuseum.org/hamsmith/McKay.html The McKay Correspondence], Tony Smith
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| * [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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| {{DEFAULTSORT:Ade Classification}}
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| [[Category:Lie groups]]
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