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A note on a cluster structure of the coordinate ring of a simple algebraic group

Hironori Oya

Abstract

We show that the coordinate ring of a simply-connected simple algebraic group $G$ over the complex number field coincides with the Berenstein--Fomin--Zelevinsky cluster algebra and its upper cluster algebra, at least when $G$ is not of type $F_4$.

A note on a cluster structure of the coordinate ring of a simple algebraic group

Abstract

We show that the coordinate ring of a simply-connected simple algebraic group over the complex number field coincides with the Berenstein--Fomin--Zelevinsky cluster algebra and its upper cluster algebra, at least when is not of type .

Paper Structure

This paper contains 8 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Simple algebraic groups
  4. Cluster algebras
  5. Berenstein--Fomin--Zelevinsky seeds
  6. Main result
  7. Main result
  8. Proof of \ref{['t:mainminor']} in the cases of type $E_8$ and $G_2$

Key Result

Theorem 1

Let $G$ be a simply-connected simple algebraic group over $\mathbb{C}$ which is not of type $F_4$. Then where $\mathscr{A}$ is the cluster algebra associated with the Berenstein--Fomin--Zelevinsky seed for the open double Bruhat cell in $G$, and $\mathscr{U}$ is the corresponding upper cluster algebra. Here we do not add the inverse of the frozen variables in the definition of $\mathscr{A}$ and $

Theorems & Definitions (24)

  • Theorem : =\ref{['t:main']}
  • Theorem 2.1: GLS13
  • Lemma 2.2
  • proof
  • Theorem 2.3: CAIII, SW21
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 14 more