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Cluster algebras and Poisson geometry. (English) Zbl 1057.53064
Roughly speaking, cluster algebras, introduced recently by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], are defined by \(n\)-regular trees whose vertices correspond to \(n\)-tuples of cluster variables and edges describing birational transformations between two \(n\)-tuples of variables. Model examples of cluster algebras are coordinate rings of double Bruhat cells. Also every \(m\times n\), \(m\leq n\), integer matrix \(Z\), such that the matrix \([(DZ)_{ij}]_{i,j\leq m}\) is skew-symmetric for a certain positive integer diagonal \(m\times m\) matrix \(D\), defines a natural cluster algebra \({\mathcal A}(Z)\) of geometric type.
The authors introduce in the paper the cluster manifold \(\mathcal X\) as a “handy” nonsingular part of \(\text{ Spec}({\mathcal A}(Z))\) and a Poisson structure on \(\mathcal X\) which is compatible with the cluster algebra structure in the sense that the Poisson bracket is homogeneously quadratic in any set of cluster variables. Then edge transformations describe simply transvections with respect to the Poisson structure. Poisson and topological properties of the union of generic orbits of a toric action on this Poisson variety are studied.
The second goal of the paper is to extend calculations of the number of connected components in double Bruhat cells to a more general setting of geometric cluster algebras and compatible Poisson structures. Namely, given a cluster algebra \(\mathcal A\) over the reals, the number of connected components in the union of generic symplectic leaves of any compatible Poisson structure on \(\mathcal X\) is computed. Finally, the general formula is applied to a special case of Grassmannian coordinate ring.

53D17 Poisson manifolds; Poisson groupoids and algebroids
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
16S99 Associative rings and algebras arising under various constructions
17B20 Simple, semisimple, reductive (super)algebras
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