Periodicities in cluster algebras and dilogarithm identities. (English) Zbl 1320.13029
Skowroński, Andrzej (ed.) et al., Representations of algebras and related topics. Proceedings of the 14th international conference on representations of algebras and workshop (ICRA XIV), Tokyo, Japan, August 6–15, 2010. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-101-9/hbk). EMS Series of Congress Reports, 407-443 (2011).
From the introduction: Cluster algebras were introduced by Fomin and Zelevinsky. They naturally appear in several different areas of mathematics, for example, in geometry of surfaces, in coordinate rings of algebraic varieties related to Lie groups, in the representation theory of algebras, and also in the representation theory of quantum groups, etc..
In this paper we focus on two kinds of periodicities of mutations in general cluster algebras. The first one is the periodicity of exchange matrices (or quivers) under a sequence of mutations. In other words, the exchange relations of clusters and coefficient tuples are periodic under such a sequence of mutations. The second one is the periodicity of seeds under a sequence of mutations. The latter periodicity implies the former one, but the converse is not true.
For the entire collection see [Zbl 1226.16001].
In this paper we focus on two kinds of periodicities of mutations in general cluster algebras. The first one is the periodicity of exchange matrices (or quivers) under a sequence of mutations. In other words, the exchange relations of clusters and coefficient tuples are periodic under such a sequence of mutations. The second one is the periodicity of seeds under a sequence of mutations. The latter periodicity implies the former one, but the converse is not true.
For the entire collection see [Zbl 1226.16001].
MSC:
| 13F60 | Cluster algebras |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 33B30 | Higher logarithm functions |
