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Exhange graphs for mutation-finite non-integer quivers

Felikson, Anna; Lampe, Philipp

Exhange graphs for mutation-finite non-integer quivers Thumbnail


Philipp Lampe


Skew-symmetric non-integer matrices with real entries can be viewed as quivers with noninteger arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).


Felikson, A., & Lampe, P. (2023). Exhange graphs for mutation-finite non-integer quivers. Journal of Geometry and Physics, 188, Article 104811.

Journal Article Type Article
Acceptance Date Mar 23, 2023
Online Publication Date Apr 6, 2023
Publication Date 2023
Deposit Date Mar 30, 2023
Publicly Available Date May 10, 2023
Journal Journal of Geometry and Physics
Print ISSN 0393-0440
Electronic ISSN 1879-1662
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 188
Article Number 104811
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