Twist automorphism for a generalized root system of affine ADE type

TL;DR

This paper introduces a twist automorphism for generalized root systems of affine ADE type, establishing isomorphisms with extended affine Weyl groups and exploring related algebraic structures and their geometric implications.

Contribution

It defines a new twist automorphism for affine ADE root systems and proves its role in isomorphisms with extended affine Weyl groups and related algebraic groups.

Findings

The extended affine Weyl group is isomorphic to the modified group incorporating the twist automorphism.

The number of root bases with a Coxeter transformation modulo the twist equals the degree of the Lyashko-Looijenga map.

Relationships between extended affine Weyl groups and extended Seidel-Thomas braid groups are established.

Abstract

For a generalized root system of affine ADE type, we introduce a twist automorphism. We prove that the Dubrovin-Zhang extended affine Weyl group is isomorphic to our (modified) extended affine Weyl group, which is an extension of the affine Weyl group by the twist automorphism. We also show that the number of root bases with a Coxeter transformation modulo the twist automorphism is equal to the degree of the Lyashko-Looijenga map of the Frobenius manifold constructed by Dubrovin-Zhang. As analogues of the extended affine Weyl group, we define an extended Artin group and an extended Seidel-Thomas braid group. We study the relationship between the extended affine Weyl group and the extended Seidel-Thomas braid group.