Rigidity of quadratic Poisson tori

TL;DR

This paper establishes a rigidity theorem for Poisson automorphisms of quadratic Poisson tori, enabling classification of automorphism groups in related algebraic structures and confirming a Poisson analogue of a conjecture on quantum groups.

Contribution

It introduces a rigidity theorem for Poisson automorphisms of quadratic Poisson tori and applies it to classify automorphism groups of cluster algebras and flag varieties.

Findings

Rigidity theorem for Poisson automorphisms of quadratic Poisson tori.

Classification of automorphism groups of cluster algebras with Gekhtman-Shapiro-Vainshtein structures.

Confirmation of a Poisson version of the Andruskiewitsch-Dumas conjecture.

Abstract

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coordinate rings can be identified with the semiclassical limits of the positive parts $U_q({\mathfrak{n}}_+)$ of the quantized universal enveloping algebras of semisimple Lie algebras, and the last result establishes a Poisson version of the Andruskiewitsch-Dumas conjecture on ${\mathrm{Aut}} U_q({\mathfrak{n}}_+)$.