Folding Map of Dynkin Diagrams

• Folding maps of Dynkin diagrams are algebraic and geometric transformations that utilize finite group automorphisms to produce lower-rank or twisted diagrams.
• The method decomposes the root lattice into invariant and orthogonal sublattices, enabling classification of new symmetry structures and dynamics.
• These techniques underpin the classification of q-Painlevé equations and integrable systems by inducing non-birational algebraic transformations and moduli space reductions.

A folding map of Dynkin diagrams is an algebraic and geometric operation that systematically transforms a Dynkin diagram—encoding the root system, symmetry, or classification data of Lie algebras, algebraic surfaces, or integrable systems—into another diagram, typically of lower rank or with more intricate edge structure. The folding is implemented via diagram automorphisms or their group actions, yielding new root systems, modified symmetries, or induced transformations between moduli spaces and differential or difference equations. Folding maps play a foundational role in the representation theory of algebraic groups, the structure theory of Kac–Moody and affine Lie algebras, the classification of Painlevé equations, and the dynamics of algebraic and geometric systems with symmetry.

1. Algebraic Definition and Lattice Decomposition

The folding map is formally realized via the action of a finite group of automorphisms (typically a cyclic group $C$) on a Dynkin diagram or its associated root lattice. Let $W^{(ae)}$ denote an extended affine Weyl group (the symmetry group acting on the Picard lattice $Q^a$ of a rational surface). A folding transformation corresponds to a finite order element $w \in W^{(ae)}$ generating a subgroup $\langle w \rangle$. The lattice $Q^a$ decomposes orthogonally (over $\mathbb{Q}$) into two key sublattices:

• The invariant lattice $Q^{(a,w)} = \mathrm{Ker}(w-1)$, containing elements fixed by $w$ and typically encompassing the imaginary root $\delta$.
• The orthogonal lattice $$Q^{(a,\perp w)} = \mathrm{Ker}\left( \sum_{i=0}^{n-1} w^i \right )$$, consisting of elements that average to zero under the $w$-action.

These two sublattices are orthogonal, and their direct sum yields a full-rank subgroup of $Q^a$ over $\mathbb{Q}$. The structure of $Q^{(a,w)}$ captures the essence of the folded symmetry: it often forms a root lattice of a new (possibly non-simply-laced or twisted) root system, with norm squared possibly distinctly scaled (e.g., $|\alpha|^2 = 2n$ for "long" roots in $A_2^{(1)}$ after an $E_8^{(1)} \to E_6^{(1)}$ folding) (Bershtein et al., 2021).

2. Folding Transformations and Dynkin Diagram Automorphisms

Folding transformations for $q$-Painlevé and related equations correspond to automorphisms of affine Dynkin diagrams or, more generally, to certain symmetry subdiagrams. Algebraically, an automorphism $\sigma$ of a Dynkin diagram permutes its nodes, and the identification of nodes under $\sigma$ yields a new, typically non-simply-laced Dynkin diagram. The root system of the quotient (folded) diagram can be realized as the fixed subspace of the root lattice under $\sigma$. For each orbit $I$ of the node set under $\sigma$, the composite (folded) simple root $\bar{\alpha}_I = \sum_{i \in I} \alpha_i$ defines the nodes of the new diagram, and the Cartan matrix of the folded diagram is determined by

$$\bar{A}_{IJ} = \sum_{i \in I} a_{ij},\ \text{where }j \in J$$

with $a_{ij}$ the entries of the original Cartan matrix (Liu et al., 2018).

In the context of the space of initial conditions of $q$-Painlevé equations (typically rational surfaces with symmetry given by an affine Weyl group), such automorphisms act nontrivially on the moduli but leave root parameters invariant, producing algebraic transformations ("foldings") of the equations that are not birational but of higher degree (Bershtein et al., 2021).

3. Centralizer, Normalizer, and Symmetry Reduction

A folding transformation $w$ produces not only a new symmetry lattice but also modifies the algebraic and geometric symmetry structure of the system. The relevant subgroup is the centralizer $$C(w, W^{(ae)}) = \{ g \in W^{(ae)} \mid g w = w g \}$$, implementing symmetries compatible with $w$. Additionally, the normalizer $$N(\langle w \rangle, W^{(ae)}) = N_\mathrm{flip} \ltimes C(w, W^{(ae)})$$ can contain flip symmetries when $\langle w \rangle$ has several conjugate generators.

The centralizer $C(w, W^{(ae)})$ typically acts on the invariant lattice $Q^{(a,w)}$, and the induced Weyl group $W^{(a)}_{(\Phi^{(a,w)})}$ associated to the root system with this lattice as its root lattice emerges as the symmetry group of the folded system. In numerous examples, the dynamics after folding—such as the translation part of the $q$-Painlevé difference operator—are governed by this folded affine Weyl group or a finite extension thereof (Bershtein et al., 2021).

If the folding introduces projective reductions (i.e., extraction of roots of translations for certain parameter specializations), fractional dynamics such as $t^{1/n}_\lambda$ can appear, corresponding to further symmetry reductions (Bershtein et al., 2021).

4. Geometric Realization via Rational Surfaces

The geometric incarnation of folding arises in the context of rational surfaces associated with Painlevé equations, notably within Sakai's classification. A rational surface $X$ with symmetry group $W^{(ae)}$ (the extended affine Weyl group) admits a cyclic group action induced by $w$. The quotient surface $Y = X / \langle w \rangle$ (after resolving singularities and blowing down $(-1)$-curves when necessary) inherits a symmetry described by the folded Dynkin diagram.

Intersection-theoretic formulas detail how divisors descend to the quotient:

$$\tilde{D} \cdot \tilde{D} = \frac{1}{m} (C \cdot C) - \sum_i \frac{d_i}{m_i}$$

where $m$ is the order of $w$, $m_i$ the size of the stabilizer for each divisor, and the sum is over exceptional curves (Bershtein et al., 2021).

Nodal curves (corresponding to particular root variables taking value $1$) generate a sublattice $Q_{nod}$; their orthogonal complement in $Q^a$ governs the symmetry of the quotient and the effective symmetry lattice of the folded Painlevé equation.

5. Classification and Explicit Folds in the Painlevé Context

Folding transformations for the $q$-Painlevé equations are completely classified by their correspondence with automorphisms/subdiagrams of affine Dynkin diagrams (Bershtein et al., 2021). The method proceeds via:

• Enumerating elements $w$ of finite order in $W^{(ae)}$ (typically via diagram automorphisms);
• Decomposing the lattice $Q^a$ into $Q^{(a,w)}$ and $Q^{(a,\perp w)}$;
• Identifying the induced symmetry (by centralizer computation), mapping to affine Weyl groups of the folded diagram;
• Analyzing quotient geometries, singularities, and possible projective reductions.

In concrete cases such as the $E_8^{(1)} \rightarrow E_6^{(1)}$ folding, the invariant lattice becomes the root lattice of $A_2^{(1)}$ with appropriately rescaled root lengths, and the dynamics of the $q$-Painlevé equation after folding is governed by translations/extensions in the centralizer of $w$ (Bershtein et al., 2021).

6. Dynamical and Moduli-Theoretic Consequences

The effect of folding on the Painlevé (and $q$-Painlevé) equations is to relate solutions of different equations by non-birational, algebraic transformations (of degree $>1$). These transformations map the space of initial conditions (moduli space) of one equation to another, identifying invariant configurations under $w$ (or its action on parameters) and modifying the dynamics accordingly. Such foldings explain and classify algebraic reductions, symmetries, and degenerations between families of Painlevé-type equations and encode the connection between geometrically distinct but algebraically intertwined moduli spaces (Bershtein et al., 2021).

7. Summary Table: Lattice Decomposition Under Folding

Sub-Lattice	Description	Algebraic Role
$Q^{(a,w)}$	Invariant sublattice under $w$	Folded root lattice, dynamics
$Q^{(a, \perp w)}$	Averaging-to-zero sublattice under $w$	Orthogonal complement, symmetry
$Q_{nod}$	Nodal-curve sublattice (for roots = $1$)	Defines trivial reflections

This algebraic-geometric synthesis of folding transformations situates the process as a bridge between group-theoretic automorphism, surface geometry, and symmetry reduction in integrable and moduli-theoretic contexts. It unifies the classification of $q$-Painlevé and related equations under algebraic transformations with the theory of Dynkin diagrams and affine Weyl groups (Bershtein et al., 2021).