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

Abstract

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