Cluster algebras and Poisson geometry by Michael Gekhtman

By Michael Gekhtman

Cluster algebras, brought through Fomin and Zelevinsky in 2001, are commutative jewelry with unit and no 0 divisors built with a distinct kin of turbines (cluster variables) grouped in overlapping subsets (clusters) of a similar cardinality (the rank of the cluster algebra) attached by means of alternate kin. Examples of cluster algebras contain coordinate jewelry of many algebraic kinds that play a favourite position in illustration concept, invariant idea, the examine of overall positivity, and so on. the idea of cluster algebras has witnessed a excellent development, at the start a result of many hyperlinks to a variety of topics together with illustration idea, discrete dynamical structures, Teichmüller conception, and commutative and non-commutative algebraic geometry. This e-book is the 1st dedicated to cluster algebras. After featuring the required introductory fabric approximately Poisson geometry and Schubert types within the first chapters, the authors introduce cluster algebras and end up their major houses in bankruptcy three. This bankruptcy might be seen as a primer at the thought of cluster algebras. within the ultimate chapters, the emphasis is made on geometric elements of the cluster algebra thought, specifically on its family to Poisson geometry and to the idea of integrable platforms

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Mb , where b is the number of bounded indices in i. 5). • The generators of O(Lu,v ) are organized into b + 1 groups (“clusters”) numbered 0 to b. Group 0 corresponds to the toric chart Ui , group q is associated with the toric chart Uq,i . Cluster 0 contains M1 , . . , Mb , cluster q contains M1 , . . , Mq−1 , Mq , Mq+1 , . . , Mb . These clusters form a star-like graph with b + 1 labeled vertices and edges connecting vertex 0 with all other vertices (see Fig. 19). 5). 19. Star on 7 clusters An algebra with such a special structure of generators is an example of the so called upper cluster algebra.

For the generalization of the same approach to double Bruhat cells, see [SSVZ, Z, GSV1]. Double Bruhat cells for semisimple Lie groups are introduced in [FZ1], reduced double Bruhat cells and the inverse problem of restoring factorization parameters are studied in [BZ2]. 3 we follow [SSVZ, Z]. 11 is proved in [Z]. The construction of functions Mi in the SLn -case based on pseudoline arrangements is borrowed from [BFZ1, FZ1]. 14 are obtained in [FZ1]. Functions τi are introduced and studied in [SSV2].

Mm . 34 2. BASIC EXAMPLES: RINGS OF FUNCTIONS ON SCHUBERT VARIETIES We start with the irreducibility of Mn . Assume that Mn = P · Q, where P and Q are some regular functions on Lu,v . Restrict all functions to Ui . Since P and Q are regular functions on Ui , they are Laurent polynomials in M1 , . . , Mm . Equality Mn = P ·Q implies that both P and Q are in fact monomials. However, a monomial dm is regular on Lu,v if and only if all di corresponding to i-bounded i M1d1 ×· · · ×Mm dm must be a regular function are nonnegative.

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