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Twist automorphism for a generalized root system of affine ADE type

Takumi Otani

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.

Twist automorphism for a generalized root system of affine ADE type

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.

Paper Structure

This paper contains 14 sections, 38 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized roost system
  3. Root system
  4. Generalized root system
  5. Twist automorphism and extended affine Weyl group
  6. Generalized root system of affine ADE type
  7. Twist automorphism
  8. Extended affine Weyl groups
  9. Twist automorphism and spherical twist functor
  10. Derived category and generalized affine root system
  11. Lyashko--Looijenga map and root bases
  12. Extended Artin group and spherical twists
  13. Extended Artin group
  14. Extended Seidel--Thomas braid group

Key Result

Theorem 1.1

There is a group isomorphism $\widehat{W}_A \cong \widehat{W}_A^\mathrm{DZ}$. Moreover, there exists a $\widehat{W}_A$-equivariant biholomorphism $\widehat{\varphi} \colon {\mathcal{X}}_A \longrightarrow \widehat{{\mathfrak h}}_A$.

Figures (3)

  • Figure 1: Dynkin diagram $\Gamma_A$
  • Figure 2: The Coxeter--Dynkin diagram $\widetilde{\Gamma}_A$
  • Figure 3: Quiver $Q_{\widetilde{{\mathbb T}}_A}$

Theorems & Definitions (75)

  • Theorem 1.1: Proposition \ref{['prop : extended affine Weyl groups']} and Theorem \ref{['thm : main 1']}
  • Theorem 1.2: Corollary \ref{['cor : LL map and root bases']}
  • Theorem 1.3: Theorem \ref{['thm : extended Artin group and extended ST group']}
  • Definition 2.1: S1S3
  • Definition 2.2: NST
  • Lemma 2.3: NST
  • Lemma 3.1: cf. NST
  • Proof
  • Definition 3.2
  • Remark 3.3
  • ...and 65 more