Coxeter Wheel in ADE Lie Algebras

• Coxeter wheel is a geometric and algebraic construct that represents the root systems of ADE Lie algebras using cyclic polygonal diagrams.
• The model translates Milnor fiber topology into an explicit correspondence between oriented arcs and algebraic roots, reproducing Dynkin combinatorics and Cartan matrix relations.
• Its rotational symmetry encapsulates the action of the Coxeter element, offering new insights into reflection groups, monodromy, and singularity theory.

The Coxeter wheel is a geometric and algebraic object that encodes the root system structure of simply-laced simple Lie algebras of ADE type (Aₙ, Dₙ, E₆, E₇, E₈) using a polygonal diagram with cyclic symmetry. Derived from the Milnor fiber of the corresponding simple singularity, the Coxeter wheel provides a fully geometric realization of the root system, including an explicit dictionary between oriented edges/spokes of the wheel and the roots, as well as a presentation of the Lie algebra relations via intersection and monodromy data. The construction subsumes the combinatorics of Dynkin diagrams, the actions of the Weyl group and Coxeter element, and the symplectic topology of singularity theory.

1. Construction and Geometric Model

Given an ADE singularity, the Coxeter wheel is constructed as a polygonal configuration in the plane, with vertices and edges derived from the Milnor fiber of the singularity. For Aₙ, the wheel is an (n+1)-gon; for Dₙ and Eₙ types, more elaborate coverings and unfoldings are required to realize the full cyclic symmetry corresponding to the Coxeter element.

• Edges and spokes of the Coxeter wheel are oriented arcs (line segments) connecting pairs of distinguished points. Each arc represents a “potential root”, and orientations are fixed according to a canonical labeling derived from the geometry of the Milnor fiber.
• The arrangement of edges and spokes imparts a full cyclic symmetry to the wheel, geometrically realizing the action of the Coxeter element as a rotation by $2\pi/h$, where $h$ is the Coxeter number for the corresponding type.
• Significance: The polygonal model not only encodes the combinatorics of the root system but provides explicit geometric operations—“flips” of arcs—that correspond to Weyl group reflections.

2. Geometric Root System and Equivalence of Arcs

To capture the root system algebraically, an equivalence relation is defined on the set of arcs:

• Two arcs are equivalent if and only if they represent the same relative homology class in the Milnor fiber.
• The set of equivalence classes of these arcs constitutes a geometric root system $\Phi$ of ADE type.
• The key structural result is that the intersection pairing (given by the negative of the symmetrized Seifert form from the Milnor fiber) matches the Cartan matrix of the corresponding Lie algebra:

$$(\alpha, \alpha) = 2, \quad (\alpha, \beta) = -1 \ \text{if the arcs meet appropriately}, \quad (\alpha, \beta) = 0 \ \text{otherwise}.$$

• Thus, every root in the classical Cartan–Killing root system arises canonically as an equivalence class of arcs/spokes, and the root combinatorics are realized geometrically within the wheel.

3. Milnor Fiber, Seifert Form, and Monodromy

The Coxeter wheel is not abstract: it is rooted in the symplectic topology of singularities.

• The starting point is a Milnor fiber $M$ associated to an isolated plane curve singularity (e.g., $x^2 + y^{n+1} = 0$ for type Aₙ).
• A basis of arcs in $M$ is selected so that their intersection and variation (under the monodromy operator) reproduce the structure of the Dynkin diagram.
• The Seifert form $\mathcal{L}$ on the relative homology $H_1(M, \partial M)$ encodes linking of vanishing cycles and, after symmetrization, gives rise to the Cartan matrix:

$$(\alpha, \beta) = \mathcal{L}(\alpha, \beta) + \mathcal{L}(\beta, \alpha)$$

for root classes $\alpha, \beta$.

• The variation operator (the failure of the identity under monodromy) determines the Cartan subalgebra action.

4. Geometric Model of the Simple Lie Algebra

The Coxeter wheel provides a precise geometric presentation of the Lie algebra:

• Generators are attached to each geometric root (equivalence class of oriented arc): $g_\alpha$ and $g_{-\alpha}$.
• Cartan subalgebra is modeled on the homology $H_1(M;\mathbb{Z})$, with basis elements given by a distinguished set of arcs on the wheel. The action of Cartan elements on root generators is via

$[h, g_\alpha] = \alpha(h) g_\alpha$

for $h \in H_1(M;\mathbb{Z})$.

• Root system relations are encoded by the intersection and variation data:

$$[g_\alpha, g_{-\alpha}] = -\mathrm{var}(\alpha), \quad [g_\alpha, g_\beta] = \begin{cases} N_{\alpha,\beta} g_{\alpha+\beta} & \text{if } \alpha+\beta \in \Phi \ 0 & \text{otherwise} \end{cases}$$

where $N_{\alpha,\beta}$ is a sign determined by intersection number.

• These are precisely the Serre relations for the simply-laced simple Lie algebras.

5. Coxeter Element, Rotational Symmetry, and Monodromy

The cyclic structure of the wheel gives a direct geometric realization of the Coxeter element and its action:

• The monodromy operator $\rho_*$ on the homology of the Milnor fiber corresponds (after composing with orientation reversal) to the Coxeter element $c$:

$$c = \overline{\rho}_* = (\text{orientation reversal}) \circ \rho_*$$

• Cyclic rotation by $2\pi/h$ on the wheel matches the algebraic action of $c$ on the root system.
• This correspondence demonstrates that the dynamics of classical Lie algebra symmetries and their associated reflection groups can be modeled purely through transformations of geometric arcs in the wheel.

6. Mathematical Formulations and Summary Table

Several structural equations described above are summarized here:

Structure	Mathematical Representation	Geometric Realization
Root pairing	$$(\alpha, \beta) = \mathcal{L}(\alpha,\beta) + \mathcal{L}(\beta,\alpha)$$	Intersection of arcs / Seifert pairing
Reflection	$$s_\alpha(\beta) = \beta - 2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha$$	Flip of an edge on the wheel
Lie bracket	$[g_\alpha, g_{-\alpha}] = -\mathrm{var}(\alpha)$ etc.	Geometric composition of arcs
Coxeter element	$c = \overline{\rho}_*$	Rotation of the Coxeter wheel

7. Broader Significance

• The Coxeter wheel provides a geometric and topological interpretation of the ADE classification, unifying singularity theory, root system combinatorics, and Lie algebra structure.
• The explicit realization of root systems as collections of equivalent arcs or spokes in the wheel clarifies the reflection group structure and connects algebraic concepts directly with symplectic geometry.
• This framework offers new tools for exploring mirror symmetry, topological invariants of singularities, and might suggest further developments in geometric representation theory.

The Coxeter wheel thus serves as a bridge between the classical combinatorics of ADE Lie algebras, the topology of singularities, and the geometry of reflection groups, encoding the root system and Lie bracket entirely in terms of piecewise-linear and topological data derived from the Milnor fiber (Cho et al., 30 Jul 2025).