Generalized cluster structures related to Poisson duals of $\mathrm{SL}_n$

Abstract

We study Poisson varieties $(\mathrm{SL}_n,π_{\bar{\mathbfΓ}}^{\dagger})$ parameterized by Belavin--Drinfeld quadruples $\bar{\mathbfΓ}:=(\mathbfΓ,r_0)$ of type $A_{n-1}$ along with generalized cluster structures $\mathcal{GC}^{\dagger}(\mathbfΓ)$ in $\mathbb{C}[\mathrm{SL}_n]$ compatible with $π_{\bar{\mathbfΓ}}^{\dagger}$. The Poisson structure $π_{\bar{\mathbfΓ}}^{\dagger}$ is a pushforward of the Poisson structure $π_{\bar{\mathbfΓ}}^*$ of the Poisson dual $\mathrm{SL}_n^*$ of $(\mathrm{SL}_n,π_{\bar{\mathbfΓ}})$. We prove that the generalized upper cluster algebra of $\mathcal{GC}^{\dagger}(\mathbfΓ)$ is naturally isomorphic to $\mathbb{C}[\mathrm{SL}_n]$. Moreover, for any connected reductive complex group $G$ and a BD quadruple $(\mathbfΓ,r_0)$, we produce a Poisson birational map $\mathcal{Q}:(G,π_{(\mathbfΓ_{\text{std}},r_0)}^{\dagger})\dashrightarrow(G,π_{(\mathbfΓ,r_0)}^{\dagger})$, and when $G \in \{\mathrm{SL}_n,\mathrm{GL}_n\}$, we show that $\mathcal{Q}$ is a birational quasi-isomorphism between $\mathcal{GC}^\dagger(\mathbfΓ_{\text{std}})$ and $\mathcal{GC}^\dagger(\mathbfΓ)$. Lastly, for any pair of BD triples $\tilde{\mathbfΓ} \prec \mathbfΓ$ of type $A_{n-1}$ comparable in the natural order, we use the map $\mathcal{Q}$ to construct a birational quasi-isomorphism between $\mathcal{GC}^{\dagger}(\tilde{\mathbfΓ})$ and $\mathcal{GC}^{\dagger}(\mathbfΓ)$.