Quantum Birational Weyl Group Actions

• Quantum Birational Weyl Group Actions are non-commutative, birational symmetries on q-deformed coordinate algebras that extend classical Weyl group methods to quantum integrable systems and algebraic geometry.
• They are constructed via explicit automorphisms and cluster mutations that preserve quantum group relations and q-character invariances, ensuring the polynomial regularity of quantum τ-functions.
• Applications span modeling integrable hierarchies, Toda lattice dynamics, and quantum curves, with notable examples in E8^(1) phase spaces and topological string partition functions.

Quantum birational Weyl group actions are non-commutative, birational symmetries realized as automorphisms on non-commutative or $q$-deformed coordinate algebras, notably in the context of representation theory, integrable systems, and quantum algebraic geometry. Rooted in the classical theory of Weyl groups acting on root and weight lattices, these quantum extensions play a central organizing role in the theory of quantum groups, quantum integrable hierarchies, $q$-Painlevé equations, and quantum curves.

1. Birational and Quantum Weyl Group Symmetries

The classical Weyl group $W$ of a semisimple Lie algebra acts linearly on the root and weight lattices and, via birational transformations, on commutative Poisson algebras associated with integrable hierarchies and cluster algebras. The quantization of these symmetries involves constructing non-commutative (quantum) analogues—typically automorphisms of quantum tori or difference operator algebras—where the relations mimic those of the quantum group $U_q(\mathfrak{g})$ or its associated nilpotent subalgebras.

For a symmetrizable generalized Cartan matrix (GCM) $(a_{ij})$, the classical Poisson algebra is

$$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$

with a specific Poisson bracket and nilpotency enforced, as developed by Noumi–Yamada. The birational Weyl group action extends the linear Weyl action to $R$ by conjugation with Poisson flows. Explicitly, for $s_i \in W$,

$$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$

yields a birational, non-linear automorphism on $R$ (Kuroki, 2012).

The quantum theory replaces $q$0 with a quantum algebra, e.g., a skew field generated by $q$1 subject to the quantum Serre relations, together with parameter variables and difference operators. The quantum birational Weyl group automorphism associated to $q$2 is given by

$q$3

where $q$4 acts linearly on parameter variables, and the adjoint action encapsulates the $q$5-commutative structure. These $q$6 obey the Coxeter relations and provide an explicit quantum realization, even at the level of $q$7-difference operators, rather than just commutative polynomials (Kuroki, 2012, Moriyama et al., 2021).

2. Cluster Algebra and Quiver Formalism

The cluster algebra framework, introduced to encode deep combinatorial and Poisson geometric properties underlying algebraic and quantum integrable systems, is intimately connected to birational Weyl group actions. For a simple Lie algebra $q$8 with index set $q$9 and Cartan data $W$0, an infinite quiver $W$1 is defined on the vertex set $W$2, with arrows determined by the Cartan data:

• Arrows $W$3 for all $W$4
• Additional arrows based on $W$5 and its symmetrizer.

For a positive integer $W$6, the periodic quiver $W$7 is obtained by identifying vertex indices modulo $W$8 (where $W$9 is the maximal symmetrizer entry), leading to cyclic "circles" on the finite vertex set $U_q(\mathfrak{g})$0 (Inoue, 2020).

Cluster variables $U_q(\mathfrak{g})$1, $U_q(\mathfrak{g})$2 are attached to each vertex, forming rational function fields. Birational automorphisms (mutations) and specific "circle mutations" are constructed to realize the simple Weyl group reflections $U_q(\mathfrak{g})$3:

$U_q(\mathfrak{g})$4

Each $U_q(\mathfrak{g})$5 acts by explicit birational formulas on cluster variables, satisfying $U_q(\mathfrak{g})$6 and the Coxeter braid relations, checked by both direct computation and via tropical semifield reductions.

3. Quantum $U_q(\mathfrak{g})$7-Functions and Regularity

Quantum $U_q(\mathfrak{g})$8-functions are constructed to encode the birational (and quantum birational) Weyl group actions in a form well-adapted for integrable hierarchies and quantum deformations. For each integral weight $U_q(\mathfrak{g})$9 and Weyl group element $(a_{ij})$0,

$(a_{ij})$1

where $(a_{ij})$2 is the quantum monomial and $(a_{ij})$3 is the composition of the birational automorphisms corresponding to a (reduced) expression for $(a_{ij})$4 (Kuroki, 2012).

A central result is the regularity theorem: all quantum $(a_{ij})$5-functions thus generated are polynomials (with no negative powers) in the quantum dependent variables $(a_{ij})$6 and parameters—both in the classical and $(a_{ij})$7-deformed quantum group setting. This is established via translation functors, singular vectors in Verma modules, and explicit divisibility arguments.

The quantum $(a_{ij})$8-functions satisfy quantum $(a_{ij})$9-difference Hirota–Miwa equations. For type $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$0:

$$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$1

where $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$2 is a $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$3-number and the sum is cyclic over $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$4.

4. $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$5-Characters and Invariance Properties

In the context of quantum affine algebras $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$6, the $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$7-character map associates to each finite-dimensional module $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$8 a Laurent polynomial $$R = \mathbb{C}\langle f_i,\, \alpha_j^\vee : i, j \in I \rangle$$9 in variables $R$0. The image $R$1 is described as the subring annihilated by certain screening operators.

A key result is that the quantum birational Weyl group actions constructed via cluster mutations (explicitly the $R$2) preserve the subring generated by the $R$3-character variables and their combinations, and hence leave the $R$4-character invariant:

$R$5

Even when $R$6 is a root of unity, the restricted $R$7-characters remain invariant under $R$8 in the periodic cluster setting, as shown in (Inoue, 2020).

5. Quantum Birational Actions in Non-Commutative Phase Spaces and Quantum Curves

Quantum birational Weyl group actions admit natural generalizations to non-commutative phase spaces, most dramatically in the context of the affine Weyl group of exceptional type $R$9. Here, the phase space is a skew field generated by parameters, non-commutative "coordinates" $s_i \in W$0, and ancillary variables,

$s_i \in W$1

with non-trivial $s_i \in W$2-commutation relations, notably $s_i \in W$3 (Moriyama et al., 2021). The Weyl group generators $s_i \in W$4 act as birational automorphisms—typically as adjoint-type transforms and permutation/conjugation actions on the parameters and variables. The action linearizes on "tau-sections" parametrized by divisor data:

$s_i \in W$5

with the Weyl action

$s_i \in W$6

where $s_i \in W$7 is a non-commutative quantum polynomial uniquely characterized by factorization ("non-logarithmic singularity") conditions in the $s_i \in W$8 and $s_i \in W$9 variables.

Significantly, for each Weyl-invariant class $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$0, there exists a unique quantum curve $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$1 that is invariant under all $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$2:

$$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$3

The Fredholm determinant of the resulting quantum operator computes partition functions in topological string theory, and in the classical limit, these quantum curves reduce to the Seiberg–Witten or spectral curves of integrable hierarchies.

6. Applications to Integrable Systems and Lattice Toda Hierarchies

The cluster and quantum cluster realization of Weyl group actions establishes concrete connections with the theory of integrable systems, particularly the Toda lattice hierarchy. In this framework, the $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$4-variables of the infinite quiver $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$5 are mapped to Toda $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$6-functions $$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$7:

$$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$8

The Hamiltonian flow induced by the Weyl group action coincides with the lattice Toda field equation:

$$s_i(f_j) = \exp\left(\{\alpha_i^\vee,\, \cdot \} \log f_i\right)(f_j)$$9

and the Weyl reflections act by explicit birational transformations of the $R$0-functions, maintaining the Hirota bilinear structure. This uniformizes the realization of Weyl group symmetry across cluster algebra, $R$1-characters, and integrable $R$2-hierarchies (Inoue, 2020).

7. Examples: Types $R$3 and $R$4

For $R$5, the quantum birational action reduces to explicit involutive automorphisms acting on a single generator and parameter variable, preserving the polynomial structure of $R$6-functions (Kuroki, 2012).

For $R$7, the action realizes the full affine Weyl group on a non-commutative phase space of dimension 22, with non-trivial $R$8-commutation relations. The associated quantum fundamental polynomials $R$9 admit combinatorial factorizations, and the unique $q$00-invariant quantum curve $q$01 recovers the spectral geometry of the $q$02-string and topological string partition functions (Moriyama et al., 2021).

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Quantum birational Weyl group actions thus present a robust, algebraically uniform framework for realizing Weyl symmetries in quantum deformations, cluster algebras, integrable systems, and quantum curves, with deep applications ranging from representation theory to mathematical physics. The invariance, polynomiality, and explicit realization of these actions are now fundamental features in the study of quantum integrable models and their moduli.