Cluster algebras and Weil-Petersson forms

By Michael I. Gekhtman, Michael Shapiro, A. Vainshtein. 2003

212 citations recorded by the source corpus.

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmueller space, in which case the above form coincides with the classic Weil-Petersson symplectic form.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e4e781d08acd38e153ef4f79ac5b710179112a87

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Citations in the corpus

Quivers with potentials associated to triangulated surfaces (2008, background): “After the foundational papers [10], [11] and [16] on the one hand, and [7] on the other, both surface cluster algebras and quivers with potentials have attracted the attention of several authors (e.”
Cluster algebras and triangulated surfaces. Part I: Cluster complexes (2006, methodology): “ries bij are equal to 0, 1, −1, 2, or −2. For triangulations without self-folded triangles, there is no need to introduce the map πT , and the above deﬁnition of B(T) coincides with the ones given in [9, 20]. Remark 4.2. A more concrete version of Deﬁnition 4.1 that avoids the use of the map πT is b”
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “Let us start by defining B(T ) for an ordinary ideal triangulation T without self-folded triangles; this was first done in [5, 6, 21].”
Positivity for cluster algebras from surfaces (2009, background): “ the positivity conjecture holds for A, then it also holds for Aˆ. 3. Cluster algebras arising from surfaces Building on work of Fock and Goncharov [FG1, FG2], and of Gekhtman, Shapiro and Vainshtein [GSV], Fomin, Shapiro and Thurston [FST] associated a cluster algebra to any bordered surface with m”
Higher-dimensional cluster combinatorics and representation theory (2010, background): “ll deﬁne, and in which the functions IA for A ∈#TAB#Id m function as frozen variables (in other words, they cannot be mutated). In the case that d = 1, this was shown by Gekhtman, Shapiro, and Vainshtein [GSV]. ([GSV] considers more general situations, where the polygon is replaced by other surfaces”
An invitation to higher Teichm\"uller theory (2018, methodology): “For an interpretation of the Weil-Petersson form in terms of cluster algebras see [49].”
Cluster structures on strata of flag varieties (2014, methodology): “Moreover, the cluster structure 2 of [BFZ] is compatible with the standard Poisson structure on G [KZ], [GSV2, §4.3]. A second important class of examples, ﬁrst studied by Gekhtman-Shapiro-Vainshtein [GSV1] and FockGoncharov [FG], is given by decorated Teichmuller spaces¨ of Riemann surfaces with ma”
2-frieze patterns and the cluster structure of the space of polygons (2010, unknown): “, a closed differential 2-form, see [15].”