Periodicities in cluster algebras and dilogarithm identities

By Tomoki Nakanishi. 2010

70 citations recorded by the source corpus.

We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is `regular', they are particularly natural generalizations of the known `classic' T- and Y-systems. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/07348f11a1e685f8de77bed046da4649f54ea12d

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Citations in the corpus

Introduction to Cluster Algebras (2018, unknown): “We also refer the reader to [41, 22], where periodicities of cluster algebra mutations are used to derive dilogarithm identities.”
Cluster algebras and derived categories (2012, methodology): “We refer to [112] [77] for more information on the many recent developments around this subject and to [143] for more information on the dilogarithm function.”
Donaldson-Thomas theory and cluster algebras (2010, background): “ k,i◦ = −tg i,k i= k0, tg i,k +Q(i,k0) ·tgk 0,k i6= k0, ε(0) = −, tg i,k +Q(k0,i)·tgk 0,k i6= k0, ε(0) = +. Proof. This is a consequence of (2.3) and Corollary 8.6. 9The duality has proved in [Nakb] for skewsymmetric matrices. For skewsymmetrizable matrices, it is still a conjecture. 36 8.4 g-vec”