TY - JOUR
T1 - Cluster realizations of Weyl groups and higher Teichmüller theory
AU - Inoue, Rei
AU - Ishibashi, Tsukasa
AU - Oya, Hironori
N1 - Funding Information:
The authors are grateful to Mikhail Bershtein, Pavlo Gavrylenko, Ivan Ip, Bernhard Keller, Yoshiyuki Kimura, Ian Le, Mykola Semenyakin, Gus Schrader, Sasha Shapiro and Linhui Shen for valuable discussions. T. I. would like to express his gratitude to his supervisor Nariya Kawazumi for his continuous encouragement. He also wish to thank the Université de Strasbourg, where a part of this paper was written, and Vladimir Fock for his illuminating advice and hospitality. The authors are also grateful to the anonymous referee for their careful reading and incisive comments. This work was partly done when H.O. was a postdoctoral researcher at Université Paris Diderot. He is greatly indebted to David Hernandez for his encouragement and hospitality. R. I. is supported by JSPS KAKENHI Grant Number 16H03927. T. I. is supported by JSPS KAKENHI Grant Number 18J13304 and the Program for Leading Graduate Schools, MEXT, Japan. H.O. was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.
Funding Information:
The authors are grateful to Mikhail Bershtein, Pavlo Gavrylenko, Ivan Ip, Bernhard Keller, Yoshiyuki Kimura, Ian Le, Mykola Semenyakin, Gus Schrader, Sasha Shapiro and Linhui Shen for valuable discussions. T. I. would like to express his gratitude to his supervisor Nariya Kawazumi for his continuous encouragement. He also wish to thank the Universit? de Strasbourg, where a part of this paper was written, and Vladimir Fock for his illuminating advice and hospitality. The authors are also grateful to the anonymous referee for their careful reading and incisive comments. This work was partly done when H.O. was a postdoctoral researcher at Universit? Paris Diderot. He is greatly indebted to David Hernandez for his encouragement and hospitality. R. I. is supported by JSPS KAKENHI Grant Number 16H03927. T. I. is supported by JSPS KAKENHI Grant Number 18J13304 and the Program for Leading Graduate Schools, MEXT, Japan. H.O. was supported by the European Research Council under the European Union?s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2021/7
Y1 - 2021/7
N2 - For a symmetrizable Kac–Moody Lie algebra g, we construct a family of weighted quivers Qm(g) (m≥ 2) whose cluster modular group ΓQm(g) contains the Weyl group W(g) as a subgroup. We compute explicit formulae for the corresponding cluster A- and X-transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for Qm(g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Qkh(g) (k≥ 1) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.
AB - For a symmetrizable Kac–Moody Lie algebra g, we construct a family of weighted quivers Qm(g) (m≥ 2) whose cluster modular group ΓQm(g) contains the Weyl group W(g) as a subgroup. We compute explicit formulae for the corresponding cluster A- and X-transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for Qm(g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Qkh(g) (k≥ 1) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.
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U2 - 10.1007/s00029-021-00630-9
DO - 10.1007/s00029-021-00630-9
M3 - Article
AN - SCOPUS:85106948046
SN - 1022-1824
VL - 27
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 3
M1 - 37
ER -