##
**Braids, Galois groups, and some arithmetic functions.**
*(English)*
Zbl 0757.20007

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 99-120 (1991).

[For the entire collection see Zbl 0741.00019.]

Recent studies on “some new relations among the classical objects of the title” developed by the author together with G. Anderson, R. Coleman, P. Deligne, V. G. Drinfeld and other authors are surveyed. The contents are based on the author’s lecture at the Plenary Sessions of ICM90 at Kyoto.

Let \(X_ n\) (\(n=4,5,\dots\)) be the configuration space defined as the \(n\)-times product of \(\mathbb{P}^ 1\) minus the weak diagonals modulo \(PGL(2)\)-action. Then the exterior actions of the Galois group \(G_ \mathbb{Q}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the “braid groups” \(\widehat \pi_ 1(X_ n)\) (\(n\geq 4\)) arise, which we denote \(\varphi_ n: G_ \mathbb{Q}\to\text{Aut }\widehat\pi_ 1(X_ n)\). Some compatibilities of \(\varphi_ 4\) and \(\varphi_ 5\) enable one to obtain a homomorphism of \(G_ \mathbb{Q}\) into the Grothendieck-Teichmüller group \(\widehat G T\) which is a certain specified subgroup of \(\text{Aut }\widehat \pi_ 1(X_ 4)\) introduced by Drinfeld in the context of a certain transformation group of the quasi-triangular quasi-Hopf algebra structures. When \(\widehat \pi_ 1(X_ n)\) are replaced by their pronilpotent quotients, it is possible to study the representations \(\varphi_ n\) in more detail. There is an elementwise explicit description of \(\varphi_ 4(\sigma)\) “modulo double commutator subgroup of \(\pi_ 1\)” for \(\sigma\in G_ \mathbb{Q}\) due to Anderson/Coleman/Ihara- Kaneko-Yukinari, which also gives a universal expression of Jacobi sums as the “adelic beta functions”. The determination of the Galois image of \(\varphi_ n\) is basically still open, but there are elaborate approaches independently by the author and P. Deligne by linearizing the situation via lower central series of \(\widehat \pi_ 1\). Several fundamental questions in the field are raised concretely along the contexts. Among them, Question 5.3.4(i) is now solved affirmatively by the author [“On the stable derivation algebra associated with some braid groups.” Isr. J. Math. (to appear)].

Recent studies on “some new relations among the classical objects of the title” developed by the author together with G. Anderson, R. Coleman, P. Deligne, V. G. Drinfeld and other authors are surveyed. The contents are based on the author’s lecture at the Plenary Sessions of ICM90 at Kyoto.

Let \(X_ n\) (\(n=4,5,\dots\)) be the configuration space defined as the \(n\)-times product of \(\mathbb{P}^ 1\) minus the weak diagonals modulo \(PGL(2)\)-action. Then the exterior actions of the Galois group \(G_ \mathbb{Q}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the “braid groups” \(\widehat \pi_ 1(X_ n)\) (\(n\geq 4\)) arise, which we denote \(\varphi_ n: G_ \mathbb{Q}\to\text{Aut }\widehat\pi_ 1(X_ n)\). Some compatibilities of \(\varphi_ 4\) and \(\varphi_ 5\) enable one to obtain a homomorphism of \(G_ \mathbb{Q}\) into the Grothendieck-Teichmüller group \(\widehat G T\) which is a certain specified subgroup of \(\text{Aut }\widehat \pi_ 1(X_ 4)\) introduced by Drinfeld in the context of a certain transformation group of the quasi-triangular quasi-Hopf algebra structures. When \(\widehat \pi_ 1(X_ n)\) are replaced by their pronilpotent quotients, it is possible to study the representations \(\varphi_ n\) in more detail. There is an elementwise explicit description of \(\varphi_ 4(\sigma)\) “modulo double commutator subgroup of \(\pi_ 1\)” for \(\sigma\in G_ \mathbb{Q}\) due to Anderson/Coleman/Ihara- Kaneko-Yukinari, which also gives a universal expression of Jacobi sums as the “adelic beta functions”. The determination of the Galois image of \(\varphi_ n\) is basically still open, but there are elaborate approaches independently by the author and P. Deligne by linearizing the situation via lower central series of \(\widehat \pi_ 1\). Several fundamental questions in the field are raised concretely along the contexts. Among them, Question 5.3.4(i) is now solved affirmatively by the author [“On the stable derivation algebra associated with some braid groups.” Isr. J. Math. (to appear)].

Reviewer: H.Nakamura (Tokyo)

### MSC:

20F36 | Braid groups; Artin groups |

11R32 | Galois theory |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |