Poisson bracket induced by a basis change. In Section 4.2 we apply this result to
the study of Poisson and cluster algebra structures on Grassmannians. Starting
from the Sklyanin bracket on SLn, we define the corresponding Poisson bracket
on the open Schubert cell in the Grassmannian Gk(n) and construct the cluster
algebra compatible with this Poisson bracket. It turns out that this cluster algebra
is isomorphic to the ring of regular functions on the open cell in Gk(n), see Theo-
rem 4.14. We further investigate this construction and prove that an extension of
the obtained cluster algebra is isomorphic to the homogeneous coordinate ring of
the Grassmannian, see Theorem 4.17.
The smooth part of the spectrum of a cluster algebra is called the cluster
manifold and is treated in Chapter 5. The definition of the cluster manifold is
discussed in Section 5.1. In Section 5.2 we investigate the natural Poisson toric
action on the cluster manifold and provide necessary and sufficient conditions for
the extendability of the local toric action to the global one, see Lemma 5.3. We
proceed to the enumeration of connected components of the regular locus of the toric
action in Section 5.3 and extend Theorem 2.10 to this situation, see Theorem 5.9. In
Section 5.4 we study the structure of the regular locus and prove that it is foliated
into disjoint union of generic symplectic leaves of the compatible Poisson bracket,
see Theorem 5.12. In Section 5.5 we apply these results to the enumeration of
connected components in the intersection of n Schubert cells in general position in
Gk(n), see Theorem 5.15.
Note that compatible Poisson brackets are defined in Chapter 4 only for clus-
ter algebras with the extended exchange matrix of a full rank. To overcome this
restriction, we present in Chapter 6 a dual approach based on pre-symplectic rather
than on Poisson structures. In Section 6.1 we define closed 2-forms compatible with
a cluster algebra structure and provide a complete characterization of such forms
parallel to Theorem 4.5, see Theorem 6.2. Further, we define the secondary cluster
manifold and a compatible symplectic form on it, which we call the Weil–Petersson
form associated to the cluster algebra. The reason for such a name is explained
in Section 6.2, which treats our main example, the Teichm¨ uller space. We briefly
discuss Penner coordinates on the decorated Teichm¨ uller space defined by fixing
a triangulation of a Riemann surface Σ, and observing that Ptolemy relations for
these coordinates can be considered as exchange relations. The secondary cluster
manifold for the cluster algebra A(Σ) arising in this way is the Teichm¨ uller space,
and the Weil–Petersson form associated with this cluster algebra coincides with the
classical Weil–Petersson form corresponding to Σ, see Theorem 6.6. We proceed
with providing a geometric meaning for the degrees of variables in the denominators
of Laurent polynomials expressing arbitrary cluster variables in terms of the initial
cluster, see Theorem 6.7; Proposition 2.1 can be considered as a toy version of this
result. Finally, we give a geometric description of A(Σ) in terms of triangulation
equipped with a spin, see Theorem 6.9. In Section 6.3 we derive an axiomatic ap-
proach to exchange relations. We show that the class of transformations satisfying
a number of natural conditions, including the compatibility with a closed 2-form,
is very restricted, and that exchange transformations used in cluster algebras are
simplest representatives of this class, see Theorem 6.11.
In Chapter 7 we apply the results of previous chapters to prove several conjec-
tures about exchange graphs of cluster algebras listed in Section 3.5. The depen-
dence of a general cluster algebra on the coefficients is investigated in Section 7.1.
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