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Cluster structures on triangulated non-orientable surfaces

WILSON, JONATHAN,MICHAEL (2017) Cluster structures on triangulated non-orientable surfaces. Doctoral thesis, Durham University.

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In 2002, Fomin and Zelevinsky introduced a cluster algebra; a dynamical system that has already proved to be ubiquitous within mathematics. In particular, it was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces, giving birth to quasi-cluster algebras. The finite type cluster algebras possess the remarkable property of their exchanges graphs being polytopal. We discover that the finite type quasi-cluster algebras enjoy a closely related property, namely, their exchange graphs are spherical. Revealing yet more connections we unify these two frameworks via Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable marked surfaces have an associated LP-algebra. The integration of these structures is attempted in two ways. Firstly we show that the quasi-cluster algebras of unpunctured surfaces have LP structures. Secondly, to obtain a connection for all marked surfaces, we consider laminations, forging the notion of the laminated quasi-cluster algebra. We show that each marked surface exhibits a lamination which supplies the laminated quasi-cluster algebra with an LP structure.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:cluster algebras, surfaces, triangulations, Laurent phenomenon
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2017
Copyright:Copyright of this thesis is held by the author
Deposited On:31 May 2017 12:18

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