Duke Mathematical Journal

Cluster algebras and Weil-Petersson forms

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein

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In our paper [GSV], we 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 is 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 Teichmüller space, in which case the above form coincides with the classical Weil-Petersson symplectic form.

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Duke Math. J., Volume 127, Number 2 (2005), 291-311.

First available in Project Euclid: 23 March 2005

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Zentralblatt MATH identifier

Primary: 53D17: Poisson manifolds; Poisson groupoids and algebroids
Secondary: 53D30: Symplectic structures of moduli spaces 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 14M20: Rational and unirational varieties [See also 14E08]


Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek. Cluster algebras and Weil-Petersson forms. Duke Math. J. 127 (2005), no. 2, 291--311. doi:10.1215/S0012-7094-04-12723-X. https://projecteuclid.org/euclid.dmj/1111609853

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