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2018 | OriginalPaper | Chapter

Cluster Algebras from Surfaces

Lecture Notes for the CIMPA School Mar del Plata, March 2016

Author : Ralf Schiffler

Published in: Homological Methods, Representation Theory, and Cluster Algebras

Publisher: Springer International Publishing

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Abstract

Cluster algebras were introduced by Fomin and Zelevinsky [17] in 2002. Their original motivation was coming from canonical bases in Lie Theory. Today cluster algebras are connected to various fields of mathematics, including

Combinatorics (polyhedra, frieze patterns, green sequences, snake graphs, T-paths, dimer models, triangulations of surfaces)
Representation theory of finite dimensional algebras (cluster categories, cluster-tilted algebras, preprojective algebras, tilting theory, 2-Calabi–Yau categories, invariant theory)
Poisson geometry and algebraic geometry (cluster varieties, Grassmannians, stability conditions, scattering diagrams, Poisson structures on \({{\mathrm{SL}}}(n)\))
Teichmüller theory (lambda-lengths, Penner coordinates, cluster varieties)
Knot theory (Chern–Simons invariants, volume conjecture, Legendrian knots)
Dynamical systems (frieze patterns, pentagram map, integrable systems, T-systems, sine-Gordon Y-systems)
Mathematical Physics (statistical mechanics, Donaldson–Thomas invariants, quantum dilogarithm identities, BPS particles).

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previous chapter Auslander–Reiten Theory for Finite-Dimensional Algebras

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This means that \(\oplus \) is commutative, associative, and distributive with respect to the multiplication in \(\mathbb {P}\).

A homotopy between two continuous maps \(f,g:X\rightarrow Y\) is a continuous map \(h:[0, 1]\times X\rightarrow Y\) such that \(h(0,x) =f(x)\) and \(h(1,x)=g(x)\). An isotopy is a homotopy \(h\) such that for all \(t\in [0,1]\) the map \(h(t,\text {--}):X\rightarrow h(t,X)\) is a homeomorphism. In particular an isotopy of curves cannot create self-crossings.

Ouroboros: a snake devouring its tail.

There is a choice involved here which of the two is \(P_-\) and this will make a difference later when we consider expansion formulas for cluster algebras with nontrivial coefficients. One can determine \(P_-\) as follows. If a tile \(G_j\) in the snake graph has the same orientation as the corresponding quadrilateral in the surface \(S\), then \(P_-\) contains the south and the north edges of \(G_j\) if they are boundary edges, and \(P_-\) does not contain the east or the west edge of \(G_j\).

So far, restricting to the case without punctures has been for the sake of simplicity. But now we really need to make this restriction.

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Title: Cluster Algebras from Surfaces
Author: Ralf Schiffler
Publisher: Springer International Publishing
Book: Homological Methods, Representation Theory, and Cluster Algebras
Print ISBN: 978-3-319-74584-8

Electronic ISBN: 978-3-319-74585-5

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-74585-5_3

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