Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras

Abstract

A. Bondal's symplectic groupoid of triangular bilinear forms induces a Poisson structure on the space $\mathcal{A}_n$ of $n \times n$ unipotent upper-triangular matrices. It is governed by the classical $\mathfrak{so}(n)$ reflection equation. L. Chekhov and M. Shapiro described log-canonical coordinates on this groupoid via the $\mathcal{A}_n$-quiver. We introduce a birational Weyl group action on this symplectic groupoid, generated by cluster transformations associated with certain cycles of the quiver. We prove that the Poisson subalgebra of Weyl group invariants is a finite central extension of the algebra generated by the matrix entries of $\mathcal{A}_n$. J. Song embedded the $\imath$-quantum group of type $\mathrm{AI}_n$ into the quantum cluster algebra of the $\Sigma_n$-quiver (obtained by adding frozen vertices to the $\mathcal{A}_{n+1}$-quiver). Utilizing our Weyl group action, we determine the exact image of this embedding in the classical case, proving it is Poisson isomorphic to a quotient algebra of Weyl group invariants. V. Fock and L. Chekhov defined a Poisson map $\phi_n$ from the Teichm\"uller space $\mathcal{T}_{g,s}$ into $\mathcal{A}_n$. To describe the cluster structure of $\operatorname{Im}(\phi_n)$, we apply a cluster Poisson reduction to $\mathcal{A}_n$ based on the rank condition $\operatorname{rank}(A+A^T) \le 4$, which is satisfied by all $A \in \operatorname{Im}(\phi_n)$. Although the solution set of this condition has multiple irreducible components, the Weyl group acts transitively on them, making the corresponding reductions conjugate. Thus, it suffices to determine the reduction on a single component. Finally, we show that the longest element of the Weyl group corresponds to a cluster DT-transformation on the $\mathcal{A}_{2k}$-quiver, providing a canonical basis for the cluster algebra, whereas no reddening sequence exists for odd $n$.