References provided by A.D.King
He mentions other versions of ADE series, one related to del Pezzo
surfaces (1.), and one introduced by Deligne-Vogel. He added:
I think that the del Pezzo series may be closer to what you have in mind
than the Deligne-Vogel one and that Philip Boalch's paper might be
particularly interesting.
1. A2xA1 A4 D5 E6 E7 E8 ...
arising as symmetries of del Pezzo surfaces (or at least their
intersection lattices), which probably goes back to Coble and which also
seems to appear in supergravity (cf. Vafa's `mysterious duality').
Here are some references:
- In algebraic geometry:
- Classical, now considered to go back to Coble,
but see Manin's book on "Cubic Forms" for a more modern account.
Some more recent work:
-
V. Serganova, A. Skorobogatov:
Del Pezzo surfaces and representation theory,
J. Algebra and Number Theory 1 (2007) 393-419
math.AG/0611737
-
E. Colombo, B. van Geemen, E. Looijenga:
Del Pezzo moduli via root systems
arXiv:math/0702442
-
In integrable systems:
- P. Boalch:
Quivers and difference Painleve equations
arXiv:0706.2634
-
H. Sakai:
Rational surfaces associated with affine root systems
and geometry of the Painleve equations,
Comm. Math. Phys 220 (2001) 165-229.
- In supergravity:
-
B. De Wit, J. Louis:
Supersymmetry and dualites in various dimensions.
arXiv:hep-th/9801132
-
A. Iqbal, A. Neitzke and C. Vafa:
A mysterious duality.
arXiv:hep-th/0111068
-
P. Henry-Labordere, B. Julia, L. Paulot:
Borcherds symmetries in M-theory
arXiv:hep-th/0203070
2. A1 A2 G2 D4 F4 E6 E7 E8
This sequence seems to go back to Deligne and is strictly finite because they are
all subgroups of E8.
- P. Deligne:
La serie exceptionnelle de groupes de Lie,
C. R. Acad. Sci. Paris S. I Math. 322 (1996) 321-326
- J. Landsberg and L. Manivel:
Triality, Exceptional Lie Algebras and Deligne Dimension Formulas.
Adv. Math. 171 (2002) 59-85