## Duke Mathematical Journal

### Cluster algebras and Weil-Petersson forms

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 23 March 2005

https://projecteuclid.org/euclid.dmj/1111609853

Digital Object Identifier
doi:10.1215/S0012-7094-04-12723-X

Mathematical Reviews number (MathSciNet)
MR2130414

Zentralblatt MATH identifier
1079.53124

#### Citation

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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