Michael Gekhtman

Cluster Algebras and Poisson Geometry

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.|Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

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.

The Cauchy Two-Matrix Model

We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitian matrix model is related to a hyperelliptic curve.

Elementary Toda orbits and integrable lattices

We show that key features of several important integrable lattices appear naturally in a framework of the full Toda flows. Using special symplectic leaves for these flows, we construct a family of bi-Hamiltonian integrable lattices that interpolates between the nonrelativistic and relativistic Toda lattices.

Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras

1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s variables [F] the finite non-periodic Toda lattice describes an isospectral evolution on the set of tri-diagonal matrices in sl(n). It was explicitly solved and shown to be completely integrable by Moser [Mo] . In his paper [K1], Kostant comprehensively studied the generalization of Toda lattice that evolves on the set of “tri-diagonal” elements of a semisimple Lie algebra g which also turned out to be completely integrable with Poisson commuting integrals being provided by the Chevalley invariants of the algebra. Moreover, in this paper, as well as in the works by Ol’shanetsky and Perelomov [OP], Reyman and Semenov-Tian-Shansky [RSTS1], Symes [Sy], the method of the explicit integration of the Toda equations was extended to the case when evolution takes place on the dual space of the Borel subalgebra of g . This space is foliated into symplectic leaves of different dimensions and the natural question is what can be said about the Liouville complete integrability of the Toda flows on each of these leaves. In the particular case of generic symplectic leaves in sl(n) the complete integrability was proved by Deift, Li, Nanda and Tomei [DLNT]. This paper was motivated by the work [DLNT] and its Lie algebraic interpretation proposed in [S1], [S2], [EFS]. Our main result is the following

Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices

We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family of integrable lattice equations that includes both the standard and the relativistic Toda lattices.

ON SCHUR FLOWS

For the finite Schur (dmKdV) flows, a non-local Poisson structure is introduced and shown to be linked via Backlund-Darboux transformations to linear and quadratic Poisson structures for the Toda lattice. Two different Lax representations for the Schur flows are used, one to construct Backlund-Darboux transformations and the other to solve the Cauchy problem via the trigonometric moment problem.

Poisson geometry of directed networks in an annulus

As a generalization of Postnikovs construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in arXiv: 0805.3541 induce a family of Poisson structures on rational-valued matrix functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix.

On KP generators and the geometry of the HBDE

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota bilinear difference equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a “rank one condition”, tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

Solitons and almost-intertwining matrices

We define the algebraic variety of almost intertwining matrices to be the set of triples (X,Y,Z) of n×n matrices for which XZ=YX+T for a rank one matrix T. A surprisingly simple formula is given for tau functions of the KP hierarchy in terms of such triples. The tau functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars–Schneider particle system.