ADE

Patterns in Mathematics

This book was published in the London Mathematical Society Lecture Note Series in 2025, from Cambridge University Press; 9781009335966 (hardback), 9781009335980 (paperback).

Here I will keep items related to the book or its subject, the celebrated ADE diagrams and their many occurrences in mathematics.

The diagrams and the book

Two of the authors

More ADE stories

If and when I learn about new ADE stories, I will put a note here. There is no guarantee that they do not extend to, for example, all Coxeter–Dynkin diagrams, or even that such an extension is not already in the literature.

1. BMW algebras

I found this in the thesis of Dié Gijsbers at the University of Eindhoven, written under the supervision of Arjeh Cohen and David Wales. Here is some background, which lies in the subject area of knots and links.

The braid group on n strings is generated by n−1 elements s₁, …, s_n−1 satisfying the following relations:

• the braid relations: s_is_i+1s_i = s_i+1s_is_i+1;
• if |i−j| > 1 then s_i and s_j commute.

The question whether the braid group is a linear group was open for some time, but was resolved in the affirmative by Bigelow and Krammer (independently).

The Hecke algebra is a deformation of the braid group. There is a trace function on the Hecke algebra which gives rise to the so-called HOMFLY polynomial, a famous knot invariant. (The acronym is the initial letters of its discoverers, Hoste, Oceanu, Millet, Freyd, Lickorish and Yetter.)

All this can be generalised to the ADE diagrams (and parts even to all the spherical Coxeter–Dynkin diagrams, but I will stick to ADE). The Artin group generalises the braid group by taking a generator for each vertex of the diagram, and imposing the braid relation if the two vertices are adjacent and the commuting relation otherwise. The definition of the Hecke algebra also generalises. Cohen and Wales showed that all the Artin groups of type ADE are linear.

The BMW algebra, named for its discoverers Birman and Wenzel and independently Murakami, is a further deformation of the braid group. This was extended to the other ADE types by Cohen, Gijsbers and Wales.

• S. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471–486.
• J. S. Birman and H. Wenzl, Braids, link polynomials, and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249–273.
• A. M. Cohen, D. A. H. Gijsbers and D. B. Wales, Linearity of Artin groups of finite type, J. Algebra 286 (2005), 107–153.
• A. M. Cohen, D. A. H. Gijsbers and D. B. Wales, A poset connected to Artin monoids of simply laced type, J. Comb. Theory Ser. A 113 (2006), 1646–1666.
• A. M. Cohen and D. B. Wales, Linearity of Artin groups of finite type, Israel J. Math. 131 (2002), 101–123.
• J. Hoste, A. Oceanu, K. Millet, P. Freyd, W. B. R. Lickorish and D. Yester, A new polynommial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239–246.
• D. Krammer, Braid groups are linear, Ann. Math. (2) 155 (2002), 131–156.
• J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745–758.

2. E7.5

At a workshop on Algebra, Combinatorics, Dynamics and Applications in Belfast in 2013, I learned some numerology in a talk by Dmitriy Rumynin. There is much more to it than numerology, as the talk showed; but I will stick to that.

Pierre Deligne proposed that the Lie algebras E₆, E₇ and E₈ and the other exceptional Lie algebras are not sporadic but actually part of a family. Indeed, he gave a formula, namely

−2(λ+5)(λ−6)/λ(λ−1),

which for various negative rational values of λ evaluates to the dimensions of all the exceptional Lie algebras and a few small classical ones too for good measure (A₁, A₂ and D₄). In particular, for λ=−1/2, −1/3, −1/5, its values are 78, 133, 248, the dimensions of E₆, E₇, and E₈.

Now someone pointed out that there is an obvious gap here, namely λ=−1/4, for which the formula evaluates to 190. The mythical Lie algebra of dimension 190 has been named E_7.5. Dmitriy didn't want to talk about this, but when someone asked him "Does it exist?" he replied, "It is more real than I am: it has a Wikipedia page, I don't."

Indeed, there is a Lie algebra of dimension 190; it is not simple, but is related to E₇. It is an extension of a Heisenberg algebra associated with E₇ with the algebra E₇ itself, and the numerology is

133+1+56 = 190.

You can find references on the Wikipedia page.

3. A snapshot

The Mathematisches Forschungsinstitut Oberwolfach publishes a collection of Snapshots of Modern Mathematics. Number 3/2025 is an article with the title "Five ways to spell ADE" by Dani Kaufman, with nice discussions of the solutions to 1/p+1/q+1/r > 1, the ADE Coxeter–Dynkin diagrams, kaleidoscopes and reflection groups, singularities, and cluster algebras. This is also available on the IMAGINARY platform.

4. A view from V. I. Arnold

In 1981, the Soviet mathematician V. I. Arnold wrote a little book attempting to explain catastrophe theory to a general audience. It was translated into English by R. K. Thomas and published with the title Catastrophe Theory by Springer in 1983.

At the time, catastrophe theory was somewhat controversial, having its strong proponents and equally strong opponents. It seems to me that Arnold's attitude was that catastrophe theory is a dose of low-dimensional singularity theory, obscured by a large cloud of mysticism. In the final chapter, "The Mystics of Catastrophe Theory", he writes:

The nice results of singularity theory are happily not dependent on the dark mystics of catastrophe theory. But in singularity theory, as in all mathematics, there is an element of the mysterious: mathematical objects and theories, which at first seem quite independent, turn out to be closely related.

One example of such a relationship is the enigmatic (though partly understood) A, D, E-classification. It occurs in such diverse areas of mathematics as, for example, critical points of functions, Lie algebras, categories of linear spaces, caustics, wavefronts, regular polyhedra in three-dimensional space, and the Coxeter crystallographic reflection groups.

A fitting introduction to our book!