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

TL;DR

The paper resolves the GSV conjecture for $( ext{SL}_n, obreak{π^{}_{ar{oldsymbol{ar{oldsymbol{ar{oldsymbol{oldsymbol{ar{oldsymbol{ar{}}}}}}}}}}})$ by constructing generalized cluster structures compatible with the dagger Poisson bracket for arbitrary Belavin–Drinfeld data. It introduces a novel approach based on Poisson birational quasi-isomorphisms to transfer completeness and compatibility from the trivial BD data to nontrivial cases, and proves that the generalized upper cluster algebra associated with $ ext{GC}^{
f}(oldsymbol{ar{oldsymbol{oldsymbol{ar{}}}}})$ is naturally isomorphic to $ obreak{C[ ext{SL}_n]}$. For $ ext{GL}_n$ and $ ext{SL}_n$, the paper constructs a Poisson birational map $ obreak{ ext{Q}}$ between standard and nonstandard BD data and shows that, on the level of GC structures, $ obreak{ ext{Q}}$ is a birational quasi-isomorphism; this yields a robust, uniform framework to relate BD triples and their associated cluster algebras. Furthermore, the authors develop a companion family of birational maps $ obreak{ ext{G}}$ for BD triples related by removal of BD roots, enabling an inductive proof of completeness and compatibility across BD data in type $A_{n-1}$. The results pave the way for extending complete, compatible GC structures to other Lie types and link these cluster constructions to broader algebraic and geometric structures, including potential connections to Coulomb branches and quantum group realizations.

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 $\mathcal{GC}^{\dagger}(\mathbfΓ)$.

Figures (29)

• Figure 1: The neighborhood of $\varphi_{kl}$ for $k,l \neq 1$, $k+l < n$.
• Figure 2: The neighborhood of $\varphi_{1l}$ for $2 \leq l \leq n-1$.
• Figure 3: The neighborhood of $\varphi_{k1}$ for $2 \leq k \leq n-1$.
• Figure 4: The neighborhood of $\varphi_{kl}$ for (a) $k=l=1$ and (b) $k+l=n$.
• Figure 5: The neighborhood of $h_{ij}$ for $2 \leq i \leq n-1$, $2 \leq j \leq n$, $i \leq j$.

Theorems & Definitions (187)

• Remark 1.1
• Definition 1.2
• Remark 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Definition 2.1
• Definition 2.2