Generalized cluster structure on the Drinfeld double of $GL_n$

TL;DR

This paper develops a generalized cluster structure on the Drinfeld double of $GL_n$, compatible with its Poisson bracket, and derives a related structure on $GL_n$ itself, advancing the understanding of Poisson-Lie groups.

Contribution

It introduces a new generalized cluster structure on the Drinfeld double of $GL_n$ and on $GL_n$, compatible with their respective Poisson brackets.

Findings

Constructed a generalized cluster structure on the Drinfeld double of $GL_n$.

Derived a compatible cluster structure on $GL_n$ from the double.

Enhanced the framework for studying Poisson-Lie groups and their cluster structures.

Abstract

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the dual Poisson--Lie bracket.