Weyl-group invariance of the quantum group image in the cluster realization of higher Teichmüller theory

Inoue, Ishibashi and Oya (Selecta Mathematica, 2021) realized the Weyl group $W(\mathfrak{g})$ of a symmetrizable Kac–Moody algebra inside the cluster modular group of an explicit family of quivers $Q_m(\mathfrak{g})$ and, for classical finite-type $\mathfrak{g}$, induced an action of $W(\mathfrak{g})^p$ by cluster transformations on the higher Teichmüller moduli spaces of a marked surface with $p$ punctures. Of the open problems their paper recorded, the classical ones — the coincidence of the cluster and geometric Weyl actions on the $\mathcal{A}$- and $\mathcal{X}$-spaces, and the Goncharov–Shen description of the cluster Donaldson–Thomas transformation — have been settled by Goncharov and Shen (2019). One stated expectation remains unproved: that the image of the quantum group $U_q(\mathfrak{g})$, in the Ip and Schrader–Shapiro cluster realization inside the quantum higher Teichmüller $\mathcal{X}$-space of a multi-punctured disk, lies in the $W(\mathfrak{g})$-invariants. Goncharov–Shen's own quantization proves the analogous invariance for a different embedding $\kappa$ of $U_q(\mathfrak{g})$ into the Weyl-invariant part of the quantum algebra of the once-punctured disk, and Shen (2022) proved that $\kappa$ is an isomorphism for simply-laced types; what is missing is the connection to the positive-representation embedding that the source conjecture is about. This problem asks for a proof or refutation of the source conjecture, including the precise quantum formulation it requires and its relation to the Goncharov–Shen–Shen framework.

Background

A cluster algebra is a commutative algebra generated by rational functions obtained by repeatedly applying an involutive operation called mutation to an initial seed, which consists of a quiver (an oriented graph encoding an integer exchange matrix) and a tuple of cluster variables. Mutations that return the quiver to itself act on the variables as rational automorphisms; these automorphisms form the cluster modular group $\Gamma$ of the mutation class. Fock and Goncharov attached to a marked surface $\Sigma$ (an oriented surface with punctures and marked boundary points) and a split semisimple algebraic group $G$ two moduli spaces: $\mathcal{A}_{G,\Sigma}$, parametrizing twisted decorated $G$-local systems (local systems plus choice of a decorated flag at each marked point), and $\mathcal{X}_{G',\Sigma}$, parametrizing framed local systems for the adjoint group $G'=G/Z(G)$. Both carry cluster structures and form a cluster ensemble; for $G=\mathrm{SL}_2$ these are classical Teichmüller theory and its enhanced Teichmüller space, so they are called the higher Teichmüller spaces. The Weyl group $W(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ of $G$ is the finite reflection group generated by the simple reflections $r_s$ indexed by the simple roots; $h$ denotes the Coxeter number, and $w_0$ the longest element of $W(\mathfrak{g})$.

Inoue–Ishibashi–Oya constructed, for each symmetrizable Kac–Moody algebra $\mathfrak{g}$ and each integer $m\geq 2$, a periodic weighted quiver $Q_m(\mathfrak{g})$ whose cluster modular group contains $W(\mathfrak{g})$, with explicit mutation sequences $R(s)$ representing the simple reflections. When $\mathfrak{g}$ is of classical finite type and $m=kh$ is a multiple of the Coxeter number, this quiver (with suitable frozen, i.e. non-mutating, vertices attached) is mutation-equivalent to the quiver encoding the cluster structure of the once-punctured disk with $2k$ boundary marked points — a structure known from Fock–Goncharov for type $A_n$ and from Ian Le for the other classical types. Gluing then realizes an action of the product $W(\mathfrak{g})^p$, where $p$ is the number of punctures, by cluster transformations of $\mathcal{A}_{G,\Sigma}$ and $\mathcal{X}_{G',\Sigma}$ for every admissible pair $(\Sigma,\mathfrak{g})$ (admissibility excludes a few small surfaces). Independently, Goncharov and Shen defined a geometric action of $W(\mathfrak{g})^p$ that only changes the decorations (flags) at the punctures and keeps the underlying local systems intact. Inoue–Ishibashi–Oya proved that the two actions coincide on $\mathcal{A}_{G,\Sigma}$ for classical finite types (their Theorem 3), and Goncharov–Shen subsequently generalized this to all semisimple $\mathfrak{g}$ and confirmed the $\mathcal{X}$-space counterpart (Inoue–Ishibashi–Oya's Conjecture 4). Goncharov–Shen also proved their own conjecture, recorded as Conjecture 5 in the same paper, that the cluster Donaldson–Thomas transformation of the mutation class of $\mathcal{X}_{G',\Sigma}$ — the canonical cluster transformation encoding the motivic Donaldson–Thomas invariants of the associated 3d Calabi–Yau category — is the composition $r_\Sigma\circ *\circ\mathbf{w}_0$ of the rotation of special points on each boundary component, the Dynkin involution $*\in\mathrm{Out}(G)$, and the longest Weyl element at each puncture. These developments settled all the purely classical conjectures of the paper.

The remaining question concerns quantization. The cluster $\mathcal{X}$-space carries a canonical log-canonical Poisson bracket and is quantized to a quantum torus algebra $\mathcal{O}_q(\mathcal{X})$, generated by elements $X_i$ with relations $X_iX_j=q^{\varepsilon_{ij}}X_jX_i$. Ip, and Schrader–Shapiro in type $A_n$, realized the quantum group $U_q(\mathfrak{g})$ (the quantized enveloping algebra, in fact its Langlands modular double, obtained by adjoining a second commuting copy with parameter $q^{-1}$) inside the quantum torus algebra of the $\mathcal{X}$-space of a multi-punctured disk with two boundary marked points: the images of the Chevalley generators are explicit cluster monomials, and the half-Dehn twists about punctures, realized by cluster transformations, represent the universal $R$-matrix (the quasitriangularity operator of $U_q(\mathfrak{g})$); the resulting representations are the positive representations of $U_q(\mathfrak{g})$ at split real form. Classically, Inoue–Ishibashi–Oya proved that on the $\mathcal{A}$-space of the $p$-punctured disk the half-Dehn twists $b_a$ and the Weyl generators satisfy the semidirect product relations $b_ar_s^{(a)}=r_s^{(a+1)}b_a$, $b_ar_s^{(a+1)}=r_s^{(a)}b_a$ (equation (6.4) of their paper), and that the tropical embedding of the root lattice $L(\mathfrak{g})$ into tropical $\mathcal{X}$-functions is $W(\mathfrak{g})$-equivariant (their Lemma 3.16 and Proposition 3.17); since Conjecture 4 is now confirmed, the same relation holds on the $\mathcal{X}$-space. On this basis they conjectured in Section 1.5, in one sentence, that the image of $U_q(\mathfrak{g})$ lies in the space of $W(\mathfrak{g})$-invariants — that is, that the quantized Weyl action and the embedded quantum group are compatible.

The quantum situation must be read against a closely adjacent theorem in the same literature. Goncharov–Shen's 2019 paper, which settled the classical conjectures above, also quantizes the same spaces: its Theorems 4.5 and 4.6 prove that for any split semisimple adjoint $G$ and any colored decorated surface $S$ the potential functions $W_{s,i}$ and $h$-distance functions $K_{s,i}$ at a special point $s$ admit quantum lifts that are invariant under the $W^n$-action (one Weyl group factor per puncture), and that these lifts define an embedding $\kappa_s$ of the quantum Borel algebra $U_q(\mathfrak{b})$ into the invariant subalgebra of the quantum algebra $\mathcal{O}_q(\mathcal{P}_{G,S})$ of the decorated character variety $\mathcal{P}_{G,S}$ — the framed-local-systems $\mathcal{X}$-space enhanced by a pinning at each boundary interval; when a boundary component carries exactly two special points, the full $U_q(\mathfrak{g})$ embeds into the invariants modulo setting the outer monodromy around that boundary component to $1$. Goncharov–Shen record this as the realization of a suggestion of Fock and Goncharov that $U_q(\mathfrak{g})$ be exactly the $q$-deformed algebra of $W$-invariant functions on the once-punctured disk with two special points, and Shen (2022) proved the resulting isomorphism for simply-laced $\mathfrak{g}$. These results concern Goncharov–Shen's own embedding $\kappa$, not the Ip and Schrader–Shapiro positive-representation embedding that the source's Section 1.5 conjecture is about: no work identifies the two embeddings; Goncharov–Shen moreover observe that the published proof of Ip's embedding theorem is incomplete (it assumes, without proof, that the relevant quantum cluster transformations are independent of the mutation sequence), a caveat that carries no restriction to any particular Lie type; and neither the multi-punctured statement with its braid-group and $R$-matrix structure, nor the statement without the outer-monodromy quotient, is covered. The conjecture of Inoue–Ishibashi–Oya is therefore open as stated, but any solution must now be positioned with respect to this adjacent framework rather than formulate the question from scratch.

Problem Statement

Prove or refute the Section 1.5 conjecture of Inoue–Ishibashi–Oya: in the cluster realization of the quantum group $U_q(\mathfrak{g})$ constructed by Ip (types $A_n$, $B_n$, $C_n$, $D_n$) and by Schrader–Shapiro (type $A_n$) inside the quantum algebra of the higher Teichmüller $\mathcal{X}$-space of a multi-punctured disk with two boundary marked points, the image of $U_q(\mathfrak{g})$ is contained in the subalgebra of invariants for the Weyl group action of Inoue–Ishibashi–Oya — equivalently, the quantized Weyl generators $R(s)$ commute with (act trivially on) the images of the quantum group generators.

An answer must include the following, since the source states the conjecture in a single sentence:

1. Fix the quantum formulation: specify the algebra in which the image lives (the quantum torus algebra $\mathcal{O}_q(\mathcal{X})$ of the cluster $\mathcal{X}$-structure of the punctured disk, its localization, or the Langlands modular double), and define the quantum lift of the Weyl generators $R(s)$ — the mutation sequences of Inoue–Ishibashi–Oya promoted to automorphisms of that algebra via quantum mutations — making all normalization conventions (frozen vertices, rescalings, quantum dilogarithm factors) explicit. The formulation must be faithful to the source: it must reduce, in the classical limit and in the tropical specialization, to the proven statements of the source (the relation (6.4) between half-Dehn twists and Weyl generators, and the $W(\mathfrak{g})$-equivariant tropical embedding of the root lattice). It must also be positioned explicitly against the existing quantum framework of Goncharov–Shen and Shen — the algebra $\mathcal{O}_q(\mathcal{P}_{G,S})$, its $W^n$-invariants, and the outer-monodromy quotient: either adopted from it, or justified against it, with the relation between the submitted formulation and the embedding $\kappa$ of Goncharov–Shen's Theorem 4.6 stated. The known $W^n$-invariance of the image of $\kappa$ must not be silently identified with the conjecture.

2. Establish the invariance: prove that every generator of $U_q(\mathfrak{g})$ whose image is defined by the realization is fixed by each quantized $R(s)$, for all finite types and puncture counts for which the ingredients are defined; or refute the conjecture by exhibiting an explicit element of the image of $U_q(\mathfrak{g})$ and an explicit quantized $R(s)$ that does not fix it, together with the exact computation in the quantum torus algebra. If the literal statement fails but a corrected one holds (for example only after passing to the modular double, only up to a central character, or only after identifying the Ip/Schrader–Shapiro embedding with the Goncharov–Shen embedding $\kappa$, in which case the identification must be proved), the corrected statement should be formulated and proved, with the deviation from the source statement made explicit. Since Goncharov–Shen (Section 6.2.6 of arXiv:1904.10491) record that the published proof of Theorem 4.14 of Ip (arXiv:1612.05641) — the embedding of $U_q(\mathfrak{g})$ into a quotient of the quantum torus algebra of the quiver $D^\mathfrak{g}_i$ — is incomplete, because it assumes without proof or reference that the quantum cluster transformation between the quivers of two reduced decompositions is independent of the mutation sequence, and that the proof of the crucial Theorem 5.1 of Ip's positive-representation papers (arXiv:1203.2018, arXiv:1205.2940) is incorrect, a solution that relies on Ip's embedding, for any type it covers (including but not limited to $B_n$ and $C_n$), must either supply the missing sequence-independence argument for the quantum cluster transformations involved or state the conditional status of the result explicitly.

Any restriction of scope (to simply-laced types, to a single puncture, to a specific quantization) must be stated and justified rather than silently assumed.

References

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