Introduction to Cluster Algebras

Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians \(\mathit{Gr}_{2}(\mathbb{C}^{n})\), and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.

Notes

Acknowledgements

These notes are based on lectures we presented during the 2016 ASIDE summer school in Montreal. We thank the ASIDE organizers for inviting us and also for coordinating the efforts to compile the lecture notes. We thank Sophie Morier-Genoud and the anonymous referee for several helpful comments.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

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