Book Chapter10.1007/978-3-319-74585-5_3
Cluster Algebras from Surfaces
Ralf Schiffler
- 01 Jan 2018
- pp 65-99
6
TL;DR: Cluster algebras were introduced by Fomin and Zelevinsky [17] in 2002 as discussed by the authors, and their original motivation was coming from canonical bases in Lie Theory.
read more
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 \({{\mathrm{SL}}}(n)\))
Teichmuller 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).
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Posted Content
Unistructurality of cluster algebras from surfaces without punctures
TL;DR: In this article, it was shown that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural, based on the bracelet basis and the skein relation.
3
•Posted Content
Mutation of type $D$ friezes
A. Garcia Elsener,K. Serhiyenko +1 more
TL;DR: In this article, a combinatorial formula for the entries in a frieze after mutation was provided. But the mutation formula was not applied to the sub-patterns of a sub-frieze of type A.
2
Continued Fractions in Cluster Algebras, Lattice Paths and Markov Numbers
Michelle Rabideau
- 01 Jan 2018
TL;DR: This work constructs an explicit formula for the F-polynomial of a cluster variable in a surface type cluster algebra and defines lattice paths and order them by the number of perfect matchings of their associated snake graphs.
Mutation of type D friezes
A. Garcia Elsener,K. Serhiyenko +1 more
TL;DR: A combinatorial formula for the entries in a frieze after mutation is provided to study mutation of friezes of type $D$ and the mutation formula recently found by Baur et al. is provided.
Cluster algebras and continued fractions
Ilke Canakci,Ralf Schiffler +1 more
TL;DR: In this article, the authors established a combinatorial realization of continued fractions as quotients of cardinalities of sets, which are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras.
References
Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
Cluster algebras II: Finite type classification
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, a complete classification of cluster algebras of finite type is presented, i.e., those with finitely many clusters, which is identical to the Cartan-Killing classification of semisimple Lie algebases and finite root systems.
1.1K
Tilting theory and cluster combinatorics
TL;DR: In this article, a new category C, called the cluster category, is introduced, which is obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field.
1.1K
Moduli spaces of local systems and higher Teichmüller theory
V. V. Fock,Alexander Goncharov +1 more
TL;DR: The moduli space of positive representations is a topologically trivial open domain in the space of all representations as discussed by the authors, and all positive representations of the fundamental group of S to G(R) are faithful, discrete and positive hyperbolic.
Cluster algebras IV: Coefficients
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this paper, the dependence of a cluster algebra on the choice of coefficients was studied, and it was shown that for cluster algebras with principal coefficients, the exchange graph of the cluster algebra with the same exchange matrix covers the exchange matrix of any cluster algebra of the same type.