Integrable cluster dynamics of directed networks and pentagram maps

Michael Gekhtman; Michael Shapiro; Serge Tabachnikov; Alek Vainshtein

doi:10.1016/j.aim.2016.03.023

Integrable cluster dynamics of directed networks and pentagram maps

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein

Research output: Contribution to journal › Article › peer-review

Abstract

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatibility of Poisson and cluster structures and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.

Original language	English
Pages (from-to)	390-450
Number of pages	61
Journal	Advances in Mathematics
Volume	300
DOIs	https://doi.org/10.1016/j.aim.2016.03.023
State	Published - 10 Sep 2016

Bibliographical note

Funding Information:
It is a pleasure to thank the Hausdorff Research Institute for Mathematics whose hospitality the authors enjoyed in summer of 2011 where this research was initiated. We are grateful to A. Bobenko, V. Fock, S. Fomin, M. Glick, R. Kedem, R. Kenyon, B. Khesin, G. Mari-Beffa, V. Ovsienko, R. Schwartz, F. Soloviev, Yu. Suris for stimulating discussions and to the referee for valuable comments. M.G. was partially supported by the National Science Foundation grant DMS-1101462 ; M.S. was partially supported by the National Science Foundation grants DMS-1101369 and DMS-1362352 ; S.T. was partially supported by the Simons Foundation grant No. 209361 and by the National Science Foundation grant DMS-1105442 ; A.V. was partially supported by the ISF grant No. 162/12 .

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

Cluster dynamics
Generalized pentagram maps
Liouville integrability
Mutations
Networks on surfaces

ASJC Scopus subject areas

Mathematics (all)

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10.1016/j.aim.2016.03.023

Cite this

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abstract = "The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatibility of Poisson and cluster structures and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.",

keywords = "Cluster dynamics, Generalized pentagram maps, Liouville integrability, Mutations, Networks on surfaces",

author = "Michael Gekhtman and Michael Shapiro and Serge Tabachnikov and Alek Vainshtein",

note = "Funding Information: It is a pleasure to thank the Hausdorff Research Institute for Mathematics whose hospitality the authors enjoyed in summer of 2011 where this research was initiated. We are grateful to A. Bobenko, V. Fock, S. Fomin, M. Glick, R. Kedem, R. Kenyon, B. Khesin, G. Mari-Beffa, V. Ovsienko, R. Schwartz, F. Soloviev, Yu. Suris for stimulating discussions and to the referee for valuable comments. M.G. was partially supported by the National Science Foundation grant DMS-1101462 ; M.S. was partially supported by the National Science Foundation grants DMS-1101369 and DMS-1362352 ; S.T. was partially supported by the Simons Foundation grant No. 209361 and by the National Science Foundation grant DMS-1105442 ; A.V. was partially supported by the ISF grant No. 162/12 . Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

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AU - Shapiro, Michael

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Integrable cluster dynamics of directed networks and pentagram maps

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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