Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

By Shihai Yang, Andrei Zelevinsky. 2008

78 citations recorded by the source corpus.

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan–Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0542e3679ef55f991c8d517438b78fee8906ce54

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Universal geometric cluster algebras (2012, unknown): “The definitions and results given here are inspired by the juxtaposition of several results on cluster algebras of finite type from [12,19,23,30].”
Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective (2009, background): “) Double Bruhat cells of this kind has previously appeared (for an arbitrary simple Lie group) in [23] in the context of integrable systems, and in [3] and [36] in connection with cluster algebras of finite type.”