Drinfeld double of $GL_n$ and generalized cluster structures

TL;DR

This paper constructs generalized cluster structures compatible with Poisson brackets on the Drinfeld double of $GL_n$ and on $GL_n$ itself, advancing the understanding of their algebraic and geometric properties.

Contribution

It introduces new generalized cluster structures compatible with Poisson brackets on both the Drinfeld double and $GL_n$, extending previous frameworks.

Findings

Cluster structures are compatible with Poisson brackets on the Drinfeld double.

Derived cluster structures on $GL_n$ compatible with dual Poisson-Lie group.

Provides a new algebraic framework for studying Poisson-Lie groups.

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 Poisson bracket on the dual Poisson--Lie group.