Super Caldero–Chapoton map for type A

İlke Çanakçı; Francesca Fedele; Ana Garcia Elsener; Khrystyna Serhiyenko

doi:10.1016/j.jalgebra.2025.04.018

Super Caldero–Chapoton map for type A

İlke Çanakçı
, Francesca Fedele
, Ana Garcia Elsener
, Khrystyna Serhiyenko

Mathematics

Producción científica: Article › revisión exhaustiva

Resumen

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichmüller theory to define a super version of the cluster algebra of type A and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type A.

Idioma original	English
Páginas (desde-hasta)	326-375
Número de páginas	50
Publicación	Journal of Algebra
Volumen	678
DOI	https://doi.org/10.1016/j.jalgebra.2025.04.018
Estado	Published - sept 15 2025

Nota bibliográfica

Publisher Copyright:
© 2025 The Authors

Financiación

Acknowledgments. The authors would like to thank Gregg Musiker, Nick Ovenhouse and Sylvester Zhang, as well as Pierre-guy Plamondon for helpful discussions. This work originates as part of the WINART3 Workshop at the Banff International Research Station. The authors thank BIRS and the organizers of the workshop. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Cluster algebras and representation theory” (supported by EPSRC grant no. EP/R014604/1) where the work on this paper was started. F.F. was supported by project REDCOM funded by Fondazione Cariverona - program “Ricerca Scientifica di Eccellenza 2018” and the ESPRC Programme Grant EP/W007509/1. A.G.E. was partially supported by PICT 2021-I–A-01154 Agencia Nacional de Promoción Científica y Tecnológica and by the EPSRC grant EP/R009325/1. K.S. was supported by the NSF grant DMS-2054255.

Financiadores	Número del financiador
Banff International Research Station for Mathematical Innovation and Discovery
UK Medical Research Council, Engineering and Physical Sciences Research Council	EP/R014604/1
Fondazione Cariverona - program	EP/W007509/1
Agencia Nacional de Promoción Científica y Tecnológica	EP/R009325/1
Neurosciences Foundation	DMS-2054255

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2025.04.018

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abstract = "One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{\"u}ller theory to define a super version of the cluster algebra of type A and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type A.",

keywords = "Caldero-Chapoton map, Cluster categories, Dimer covers, Snake graphs, Super algebra, Teichmuller theory",

author = "İlke {\c C}anak{\c c}ı and Francesca Fedele and \{Garcia Elsener\}, Ana and Khrystyna Serhiyenko",

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year = "2025",

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AU - Fedele, Francesca

AU - Garcia Elsener, Ana

AU - Serhiyenko, Khrystyna

PY - 2025/9/15

Y1 - 2025/9/15

N2 - One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichmüller theory to define a super version of the cluster algebra of type A and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type A.

AB - One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichmüller theory to define a super version of the cluster algebra of type A and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type A.

KW - Caldero-Chapoton map

KW - Cluster categories

KW - Dimer covers

KW - Snake graphs

KW - Super algebra

KW - Teichmuller theory

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