Generalized cluster structure on the Drinfeld double of GLnGL_n

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the dual Poisson--Lie bracket.Comment: This is a short announcement. The proofs will be published in a subsequent paper. Version 2: several misprints are correcte

Cluster algebras and Weil-Petersson forms

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.Comment: 17 pages, 7 figures; Substantial changes: new proof of Theorem 3.3. Corrected formulation and new proof of Theorem 3.4. Some other minor changes as wel