The ADE Classification: A Reference of Connected Subjects

The ADE graphs (also called simply-laced Dynkin diagrams) consist of three infinite families $A_n$, $D_n$, $E_6$, $E_7$, $E_8$. Remarkably, these same graphs classify structures across diverse areas of mathematics and physics.

Algebra

1. Simply-laced simple Lie algebras — The ADE diagrams are the Dynkin diagrams of the simple Lie algebras $\mathfrak{sl}_{n+1}$, $\mathfrak{so}_{2n}$, and the exceptional $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$.

2. Root systems — ADE graphs encode the simply-laced root systems; nodes are simple roots, edges indicate $\langle \alpha_i, \alpha_j \rangle = -1$.

3. Weyl groups — The Weyl group of each ADE Lie algebra is a finite reflection group generated by reflections through root hyperplanes.

4. Cartan matrices — The ADE graphs determine Cartan matrices $C_{ij} = 2\delta_{ij} - A_{ij}$ where $A$ is the adjacency matrix.

5. Kac-Moody algebras — Affine extensions $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_{6,7,8}$ yield affine Lie algebras; hyperbolic extensions also exist.

6. Quantum groups — Quantized universal enveloping algebras $U_q(\mathfrak{g})$ for ADE Lie algebras $\mathfrak{g}$.

7. Cluster algebras — Finite-type cluster algebras are classified by Dynkin diagrams; ADE types have finitely many cluster variables.

8. Preprojective algebras — The preprojective algebra of an ADE quiver is finite-dimensional; its representation theory connects to Kleinian singularities.

9. Nichols algebras — Certain finite-dimensional Nichols algebras are classified by ADE root systems.

Representation Theory

10. Quiver representations — Gabriel's theorem: a connected quiver has finitely many indecomposable representations iff its underlying graph is ADE.

11. Auslander-Reiten theory — The AR-quiver of the category of representations of ADE quivers has a well-understood structure with mesh relations.

12. Hereditary algebras — Path algebras of ADE quivers are hereditary algebras of finite representation type.

13. McKay correspondence — Finite subgroups $\Gamma \subset SU(2)$ correspond to ADE diagrams via the McKay quiver of irreducible representations.

14. Modular representation theory — ADE diagrams appear in the classification of blocks of group algebras.

15. Tilting theory — Tilting modules over ADE path algebras connect different derived equivalent algebras.

Finite Group Theory

16. Finite subgroups of $SU(2)$ — Cyclic $\mathbb{Z}_n$ ($A_{n-1}$), binary dihedral ($D_n$), binary tetrahedral ($E_6$), binary octahedral ($E_7$), binary icosahedral ($E_8$).

17. Finite subgroups of $SO(3)$ — Cyclic, dihedral, tetrahedral, octahedral, icosahedral — quotients of the $SU(2)$ subgroups.

18. Platonic solids — The five Platonic solids correspond to $E_6$ (tetrahedron), $E_7$ (cube/octahedron), $E_8$ (dodecahedron/icosahedron); $A_n$, $D_n$ give degenerate cases.

19. Finite Coxeter groups — ADE Weyl groups are the simply-laced finite Coxeter groups.

Algebraic Geometry

20. Du Val singularities — Surface singularities $\mathbb{C}^2/\Gamma$ for $\Gamma \subset SU(2)$ finite; resolution graphs are ADE Dynkin diagrams.

21. Kleinian singularities — Another name for du Val singularities; the exceptional divisor of minimal resolution is a tree of $\mathbb{P}^1$s with ADE intersection graph.

22. Simple surface singularities — ADE singularities are the only rational double points; defined by equations:

    • $A_n$: $x^2 + y^2 + z^{n+1} = 0$
    • $D_n$: $x^2 + y^2z + z^{n-1} = 0$
    • $E_6$: $x^2 + y^3 + z^4 = 0$
    • $E_7$: $x^2 + y^3 + yz^3 = 0$
    • $E_8$: $x^2 + y^3 + z^5 = 0$

23. Simultaneous resolution — Brieskorn-Grothendieck simultaneous resolution of ADE singularities; the base is the Cartan subalgebra quotient.

24. Springer resolution — Resolution of nilpotent cone; fibers over subregular nilpotent elements are ADE configurations of $\mathbb{P}^1$s.

25. Hilbert schemes — $\text{Hilb}^n(\mathbb{C}^2/\Gamma)$ relates to ADE representation theory via the McKay correspondence.

26. Nakajima quiver varieties — For ADE quivers, these give resolutions of Kleinian singularities and their symmetric products.

27. K3 surfaces — ADE configurations of $(-2)$-curves appear in K3 surfaces; root lattices embed in $H^2(K3, \mathbb{Z})$.

28. Del Pezzo surfaces — The Picard lattice of $dP_k$ contains root systems; $dP_8$ contains $E_8$.

29. Exceptional collections — Derived categories of del Pezzo surfaces have exceptional collections related to ADE root systems.

30. ADE fibrations — Elliptic fibrations with ADE singular fibers (Kodaira classification); important in F-theory.

Singularity Theory

31. Arnold's classification — Simple (0-modal) function singularities are classified by ADE; these are the singularities of finite Milnor number with no moduli.

32. Milnor fibration — The Milnor fiber of an ADE singularity has the homotopy type of a bouquet of spheres; the number equals the Milnor number $\mu = |$roots$|$.

33. Monodromy — The monodromy of an ADE singularity is a Coxeter element in the corresponding Weyl group.

34. Intersection form — The intersection form on vanishing cycles of ADE singularities is the corresponding Cartan matrix (up to sign).

35. Brieskorn-Pham singularities — The ADE singularities are special cases of Brieskorn-Pham singularities $\sum z_i^{a_i} = 0$.

36. Versal deformations — The base of the versal deformation of an ADE singularity is $\mathfrak{h}/W$ where $\mathfrak{h}$ is the Cartan and $W$ the Weyl group.

Combinatorics and Discrete Mathematics

37. Coxeter-Dynkin diagrams — ADE graphs are the simply-laced Coxeter-Dynkin diagrams with all edge labels equal to 3 (or unlabeled).

38. Spectral graph theory — ADE graphs are exactly the connected graphs with largest eigenvalue $< 2$ (for adjacency matrix); equivalently, the affine extensions have largest eigenvalue $= 2$.

39. Subadditive functions — ADE graphs are characterized by having positive subadditive labelings summing to the dual Coxeter number.

40. Catalan combinatorics — The number of antichains in the root poset of type $X_n$ involves Catalan-type numbers; for ADE this connects to noncrossing partitions.

41. Noncrossing partitions — The lattice of noncrossing partitions of type $W$ for ADE Weyl groups $W$ is a well-studied combinatorial object.

42. Cambrian lattices — Quotients of the weak order on ADE Weyl groups by Cambrian congruences.

43. Frieze patterns — Coxeter-Conway frieze patterns of finite type correspond to triangulations; cluster algebra connection to type $A_n$.

44. Poset theory — The root posets of ADE types have beautiful combinatorial properties; distributive lattices of order ideals.

Topology

45. Lens spaces and Seifert manifolds — Links of ADE singularities are Seifert fibered spaces; for $A_n$, these are lens spaces $L(n+1, 1)$.

46. Brieskorn spheres — The link of the $E_8$ singularity is the Poincaré homology sphere $\Sigma(2,3,5)$.

47. 3-manifold invariants — Witten-Reshetikhin-Turaev invariants for ADE gauge groups.

48. Knot theory — Torus knots $T(p,q)$ relate to $A$-type singularities; more general connections via Khovanov homology.

49. 4-manifold topology — ADE plumbing diagrams give 4-manifolds with definite intersection forms.

50. Exotic spheres — Milnor's exotic 7-spheres can be constructed as boundaries of plumbings of ADE type.

Differential Geometry

51. ALE spaces — Asymptotically locally Euclidean hyperkähler 4-manifolds are resolutions of $\mathbb{C}^2/\Gamma$ (ADE).

52. Gravitational instantons — ALE spaces are gravitational instantons; the ADE classification gives all such spaces with one end.

53. Hyperkähler quotients — Kronheimer's construction of ALE spaces as hyperkähler quotients.

54. Monopole moduli spaces — Moduli spaces of $SU(2)$ monopoles have ALE space structure.

Number Theory

55. Lattices — The ADE root lattices: $A_n \subset \mathbb{Z}^{n+1}$, $D_n \subset \mathbb{Z}^n$, $E_6, E_7, E_8$ are even unimodular (for $E_8$).

56. Theta functions — Theta functions of ADE lattices; $E_8$ theta function is a modular form.

57. Sphere packings — $E_8$ lattice gives the densest lattice packing in dimension 8; proved by Viazovska (2016).

58. Moonshine — The $E_8$ root lattice appears in monstrous moonshine connections.

59. Elliptic curves — The affine $\tilde{E}_8$ diagram appears in the classification of singular fibers of elliptic fibrations.

60. Quadratic forms — ADE root lattices give even positive-definite quadratic forms.

Mathematical Physics

61. Gauge theory — ADE groups $SU(n+1)$, $SO(2n)$, $E_6$, $E_7$, $E_8$ as gauge groups; 't Hooft anomaly matching.

62. Yang-Mills instantons — ADHM construction of instantons; ADE groups give self-dual connections.

63. Conformal field theory (2d) — ADE classification of modular invariant partition functions for $SU(2)_k$ WZW models.

64. Minimal models — The $(A, D, E)$ classification of 2d CFT minimal models.

65. Fusion categories — ADE classification of certain fusion categories and subfactors.

66. Topological field theory — 3d TFTs from ADE Chern-Simons theory.

67. Integrable systems — Toda field theories, Calogero-Moser systems for ADE root systems.

68. Solitons — ADE Toda solitons; masses proportional to components of Perron-Frobenius eigenvector.

String Theory and M-theory

69. Heterotic string — The $E_8 \times E_8$ heterotic string; anomaly cancellation requires $E_8$.

70. F-theory — ADE singular fibers in F-theory compactifications give non-Abelian gauge symmetry.

71. M-theory on ADE singularities — M-theory on $\mathbb{C}^2/\Gamma$ gives ADE gauge symmetry in 7d.

72. D-branes on singularities — D-branes on ADE singularities; the worldvolume theory has ADE gauge group.

73. ADE orbifolds — String theory on $\mathbb{C}^2/\Gamma$ orbifolds; twisted sectors correspond to conjugacy classes.

74. Geometric engineering — Engineering ADE gauge theories from Calabi-Yau singularities.

75. 6d $(2,0)$ theories — The 6d $(2,0)$ superconformal theories are classified by ADE.

76. Little string theory — Little string theories associated to ADE singularities.

77. Matrix models — ADE matrix models in string theory and M-theory.

Subfactor Theory and Operator Algebras

78. Subfactor classification — Jones's index theorem; subfactors of index $< 4$ are classified by $A_n$, $D_{2n}$, $E_6$, $E_8$.

79. Principal graphs — The principal graph of a finite-depth subfactor; ADE graphs appear at index $< 4$.

80. Planar algebras — ADE planar algebras; diagrammatic approach to subfactor theory.

81. Temperley-Lieb algebras — Quotients related to $A_n$ diagrams; Jones polynomial connection.

Category Theory and Homological Algebra

82. Derived categories — Derived categories of coherent sheaves on ADE singularity resolutions; tilting objects.

83. Triangulated categories — ADE quivers give hereditary categories with Auslander-Reiten triangles.

84. t-structures — Classification of t-structures on derived categories of ADE quiver representations.

85. Hall algebras — Hall algebras of ADE quivers recover the positive part of $U_q(\mathfrak{g})$.

86. DG categories — DG enhancements of derived categories for ADE singularities.

87. Calabi-Yau categories — ADE singularities give 2-Calabi-Yau categories.

Geometric Representation Theory

88. Springer correspondence — Relates nilpotent orbits to Weyl group representations; ADE types are simply-laced.

89. Geometric Satake — The geometric Satake correspondence for ADE groups.

90. Kazhdan-Lusztig theory — KL polynomials and cells for ADE Weyl groups.

91. Perverse sheaves — Perverse sheaves on nilpotent cones for ADE Lie algebras.

92. Character sheaves — Lusztig's character sheaves for ADE groups.

93. Symplectic resolutions — ADE singularity resolutions as symplectic resolutions; quantization.

Quantum Algebra

94. Yangians — Yangian $Y(\mathfrak{g})$ for ADE Lie algebras $\mathfrak{g}$.

95. Quantum affine algebras — $U_q(\hat{\mathfrak{g}})$ for affine ADE algebras.

96. R-matrices — Solutions of Yang-Baxter equation from ADE quantum groups.

97. Crystal bases — Kashiwara's crystal bases for ADE quantum groups.

98. Canonical bases — Lusztig's canonical bases for ADE types.

99. Categorification — Khovanov-Lauda-Rouquier algebras categorifying ADE quantum groups.

Miscellaneous

100. Adjacency spectral theory — Graphs with spectral radius $< 2$: precisely the ADE Dynkin diagrams.

101. Reflection groups — ADE Weyl groups as finite real reflection groups.

102. Braid groups — Artin braid groups of ADE type; presentations from Dynkin diagrams.

103. Hecke algebras — Iwahori-Hecke algebras for ADE Weyl groups.

104. Demazure operators — Divided difference operators for ADE root systems.

105. Schubert calculus — Schubert varieties in flag manifolds for ADE groups.

106. Invariant theory — Polynomial invariants for ADE Weyl groups acting on $\mathfrak{h}$.

107. Chevalley restriction theorem — $\mathbb{C}[\mathfrak{g}]^G \cong \mathbb{C}[\mathfrak{h}]^W$ for ADE Lie algebras.

108. Grothendieck groups — $K_0$ of representations of ADE quivers is the root lattice.

109. Picard groups — Picard group of the ADE singularity resolution is the root lattice.

110. Moment maps — Hyperkähler moment maps in Nakajima quiver variety construction.

111. Borel-Weil-Bott theorem — Line bundles and representations for ADE groups.

112. Verma modules — Verma modules for ADE Lie algebras; BGG category $\mathcal{O}$.

113. Affine Grassmannians — Affine Grassmannians for ADE groups; geometric Satake.

114. Loop groups — Loop groups $LG$ for ADE Lie groups $G$.

115. Virasoro algebra — Minimal model Virasoro representations with ADE modular invariants.

116. W-algebras — $\mathcal{W}$-algebras associated to ADE Lie algebras.

117. Vertex operator algebras — Lattice VOAs for ADE root lattices.

118. Parabolic subgroups — Classification of parabolics in ADE groups via subsets of the Dynkin diagram.

119. Bruhat order — Bruhat order on ADE Weyl groups.

120. Robinson-Schensted — RS correspondence for type $A$; generalizations to other ADE types.

121. Symmetric functions — Schur functions (type $A$); generalizations to other ADE types.

122. Macdonald polynomials — Macdonald polynomials for ADE root systems.

123. Cherednik algebras — Double affine Hecke algebras for ADE root systems.

124. Rational Cherednik algebras — Symplectic reflection algebras for ADE groups.

125. Calogero-Moser spaces — Completed phase spaces of Calogero-Moser systems for ADE root systems.

Summary Table

| Graph | Lie algebra | Finite group $\Gamma \subset SU(2)$ | Platonic dual | Singularity | |-----------|-----------------|----------------------------------------|-------------------|-----------------| | $A_n$ | $\mathfrak{sl}_{n+1}$ | Cyclic $\mathbb{Z}_{n+1}$ | (degenerate) | $x^2+y^2+z^{n+1}=0$ | | $D_n$ | $\mathfrak{so}_{2n}$ | Binary dihedral $\text{Dic}_{n-2}$ | (degenerate) | $x^2+y^2z+z^{n-1}=0$ | | $E_6$ | $\mathfrak{e}_6$ | Binary tetrahedral $2T$ | Tetrahedron | $x^2+y^3+z^4=0$ | | $E_7$ | $\mathfrak{e}_7$ | Binary octahedral $2O$ | Cube/Octahedron | $x^2+y^3+yz^3=0$ | | $E_8$ | $\mathfrak{e}_8$ | Binary icosahedral $2I$ | Dodeca/Icosahedron | $x^2+y^3+z^5=0$ |

Key References

• V.I. Arnold, Singularity Theory (Lecture notes)
• J. McKay, "Graphs, singularities, and finite groups" (1980)
• P. Slodowy, Simple Singularities and Simple Algebraic Groups (1980)
• H. Durfee, "Fifteen characterizations of rational double points and simple critical points" (1979)
• M. Hazewinkel, W. Hesselink, D. Siersma, F. Veldkamp, "The ubiquity of Coxeter-Dynkin diagrams" (1977)
• P. Gabriel, "Unzerlegbare Darstellungen I" (1972)
• E. Brieskorn, "Singular elements of semi-simple algebraic groups" (1971)