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 ADE diagrams The book

The diagrams and the book

The authors

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.

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.

It has the symmetric group S_n as a quotient obtained by adding the relations s_i² = 1.

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.

Gijsbers' thesis discusses all this in far more detail, including the connection with knots and links, the Brauer algebra, and details of the linear representations.

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.