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

Abstract

We study Poisson varieties $({\mathrm{SL}}_n,{\pi }_{\bar{\mathbf{\Gamma }}}^{†})$ parameterized by Belavin--Drinfeld quadruples $\bar{\mathbf{\Gamma }}:=(\mathbf{\Gamma },r_0)$ of type $A_{n-1}$ along with generalized cluster structures ${\mathcal{G}\mathcal{C}}^{†}(\mathbf{\Gamma })$ in $\mathbb{C}[{\mathrm{SL}}_n]$ compatible with ${\pi }_{\bar{\mathbf{\Gamma }}}^{†}$. The Poisson structure ${\pi }_{\bar{\mathbf{\Gamma }}}^{†}$ is a pushforward of the Poisson structure ${\pi }_{\bar{\mathbf{\Gamma }}}^{\ast }$ of the Poisson dual ${\mathrm{SL}}_n^{\ast }$ of $({\mathrm{SL}}_n,{\pi }_{\bar{\mathbf{\Gamma }}})$. We prove that the generalized upper cluster algebra of ${\mathcal{G}\mathcal{C}}^{†}(\mathbf{\Gamma })$ 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 $\mathcal{Q}:(G,{\pi }_{({\mathbf{\Gamma }}_{\text{std}},r_0)}^{†})⇢(G,{\pi }_{(\mathbf{\Gamma },r_0)}^{†})$, and when $G\in \{{\mathrm{SL}}_n,{\mathrm{GL}}_n\}$, we show that $\mathcal{Q}$ is a birational quasi-isomorphism between ${\mathcal{G}\mathcal{C}}^{†}({\mathbf{\Gamma }}_{\text{std}})$ and ${\mathcal{G}\mathcal{C}}^{†}(\mathbf{\Gamma })$. 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 ${\mathcal{G}\mathcal{C}}^{†}(\tilde{\mathbf{\Gamma }})$ and ${\mathcal{G}\mathcal{C}}^{†}(\mathbf{\Gamma })$.

90 pages, 29 figures