Multiple generalized cluster structures on $D(\text{GL}_n)$

TL;DR

This paper constructs a broad class of generalized cluster structures on the Drinfeld double of GL_n, aligning with Belavin-Drinfeld Poisson brackets and extending previous cluster results on GL_n.

Contribution

It introduces new generalized cluster structures on the Drinfeld double of GL_n compatible with Belavin-Drinfeld Poisson brackets, expanding the existing framework.

Findings

Constructed a large class of cluster structures on D(GL_n)

Demonstrated compatibility with Belavin-Drinfeld Poisson brackets

Extended previous cluster structures on GL_n

Abstract

We produce a large class of generalized cluster structures on the Drinfeld double of $\text{GL}_n$ that are compatible with Poisson brackets given by Belavin-Drinfeld classification. The resulting construction is compatible with the previous results on cluster structures on $\text{GL}_n$.