Geometric models of simple Lie algebras via singularity theory (2507.22836v1)

Abstract: It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.

• The paper introduces the Coxeter wheel, a polygonal model that encodes the root system by linking Milnor fiber topology to Lie algebra structure.
• It employs relative homology and a negative symmetrized Seifert form to define a geometric root system isomorphic to the classical ADE classification.
• The framework generalizes to higher dimensions and non-simply-laced types, offering new tools for categorification, symplectic geometry, and mirror symmetry.

Geometric Realizations of Simple Lie Algebras via Singularity Theory

Introduction and Motivation

The correspondence between simple Lie algebras and singularity theory, particularly through the $ADE$ classification, is a central theme in modern mathematics. The paper "Geometric models of simple Lie algebras via singularity theory" (2507.22836) develops a new geometric framework for modeling simple Lie algebras of $ADE$ type using the topology and geometry of singularities, specifically via Milnor fibers of simple curve singularities. The authors introduce the concept of the Coxeter wheel, a polygonal model that encodes the root system and Lie algebraic structure in terms of embedded arcs and their interactions, providing a direct geometric realization of the algebraic data.

Background: $ADE$ Correspondence and Singularity Theory

The $ADE$ Dynkin diagrams classify both simply-laced simple Lie algebras and simple singularities. In Lie theory, the root system and Cartan–Killing form are encoded by the Dynkin diagram, with the Cartan matrix $C$ determined by the adjacency of the diagram. In singularity theory, Arnold's classification of simple singularities also yields $ADE$ types, with the Milnor fiber of a singularity carrying rich topological and algebraic structure.

Classically, the correspondence between Lie algebras and singularities is mediated by the configuration of vanishing cycles in the Milnor fiber, whose intersection form is (up to sign) the Cartan matrix. The monodromy action on homology mirrors the Weyl group action on the root system, and the Picard–Lefschetz formula realizes reflections in the Weyl group.

Geometric Root Systems via Milnor Fibers

The authors focus on simple singularities in two variables, where the Milnor fiber $M$ is a Riemann surface with boundary. They define a geometric Cartan subalgebra as $H_1(M;\mathbb{Z})$, and the dual space $H_1(M,\partial M;\mathbb{Z})$ as the space of geometric roots. The key innovation is to define the root system not via vanishing cycles (as in the three-variable case), but via relative homology classes of embedded arcs (vanishing arcs) in $M$.

The geometric pairing on $H_1(M,\partial M;\mathbb{Z})$ is given by the negative symmetrized Seifert form:

$$(\alpha, \beta) := \mathrm{var}(\alpha) \bullet \beta + \mathrm{var}(\beta) \bullet \alpha,$$

where $\mathrm{var}$ is the variation operator and $\bullet$ denotes the intersection pairing. The set of geometric roots is then

$$\Phi_\Gamma := \{\alpha \in H_1(M,\partial M;\mathbb{Z}) \mid (\alpha, \alpha) = 2\},$$

which is shown to satisfy the axioms of a root system and is isomorphic to the classical root system of type $\Gamma$.

Coxeter Wheels: Polygonal Models for Root Systems

A central construction is the Coxeter wheel, a planar polygonal model associated to each $ADE$ type. For $A_k$, the wheel is a regular $(k+1)$-gon; for $D_k$, a $2(k-1)$-gon with a central puncture; for $E_6$, $E_7$, $E_8$, more intricate polygonal decompositions are used. The vertices correspond to punctures in the Milnor fiber, and edges and spokes (line segments between vertices or to the center) represent geometric roots.

The key result is that equivalence classes of oriented edges and spokes in the Coxeter wheel correspond bijectively to geometric roots. The geometric monodromy (arising from the weight action on the singularity) acts as a rotation of the wheel, and the orbits under this action correspond to the orbits of the Coxeter element in the Weyl group.

The construction is compatible with the Coxeter plane in Lie theory: the projections of roots onto the Coxeter plane align with the geometry of the wheel, and the action of the Coxeter element corresponds to rotation.

Geometric Construction of Lie Algebras

The authors provide a fully geometric definition of the Lie algebra structure using the data of the Milnor fiber. The Lie algebra $\mathfrak{g}_{\mathrm{geo}}$ is defined as

$$\mathfrak{g}_{\mathrm{geo}} := H_1(M;\mathbb{Z}) \oplus \bigoplus_{\alpha \in \Phi_\Gamma} \mathbb{Z} \langle g_\alpha \rangle,$$

with Lie bracket specified by:

• $[h_1, h_2] = 0$ for $h_1, h_2 \in H_1(M;\mathbb{Z})$,
• $$[h, g_\alpha] = (h \bullet \alpha + h \bullet \rho_*(\alpha)) g_\alpha$$,
• $[g_\alpha, g_{-\alpha}] = -\mathrm{var}(\alpha)$,
• $$[g_\alpha, g_\beta] = N_{\alpha,\beta} g_{\alpha+\beta}$$ if $\alpha+\beta \in \Phi_\Gamma$, $0$ otherwise, with $$N_{\alpha,\beta} = (-1)^{\mathrm{var}(\beta) \bullet \alpha}$$.

This construction recovers the classical simple Lie algebra of type $\Gamma$, with the geometric root system and Cartan–Killing form realized via the topology of the Milnor fiber and the Seifert form.

Extension to Higher Dimensions and Non-Simply-Laced Types

The construction extends to $n$-variable simple singularities via stabilization, using the Thom–Sebastiani sum and Deligne's formula for the Seifert form. The pairing is modified by a sign depending on the dimension, ensuring compatibility with the classical root system.

For non-simply-laced types ($B_k$, $C_k$, $F_4$, $G_2$), the geometric model is obtained via folding of the simply-laced root systems, corresponding to identifications of roots in the wheel model that project to the same vector in the Coxeter plane.

Weyl Group, Reflections, and Coxeter Element

The geometric model provides a combinatorial realization of the Weyl group action. Reflections correspond to flipping edges and spokes in the Coxeter wheel, and the Coxeter element is realized as the composition of monodromy and orientation reversal. The action of the Coxeter element decomposes the root system into orbits of size equal to the Coxeter number, matching Kostant's theorem.

Implications and Future Directions

This geometric framework offers a direct and visualizable model for simple Lie algebras, with potential applications in representation theory, symplectic geometry, and mirror symmetry. The explicit realization of root systems and Lie brackets in terms of arcs and their interactions in the Milnor fiber provides new tools for categorification, as in the paper of Fukaya–Seidel categories and their relation to quiver representations.

The approach suggests further avenues for generalization, including the paper of more general singularities, connections to flat metrics and translation surfaces, and deeper links with homological mirror symmetry. The geometric model may also inform computational approaches to Lie theory and singularity theory, as well as the development of new invariants in low-dimensional topology.

Conclusion

The paper establishes a comprehensive geometric model for simple Lie algebras of $ADE$ type via the topology of simple singularities, using the Coxeter wheel as a unifying structure. The identification of root systems, Lie algebraic operations, and Weyl group actions with explicit geometric data in the Milnor fiber provides a robust and versatile framework, bridging algebraic, topological, and geometric perspectives. This work opens new directions for the interplay between singularity theory, Lie theory, and symplectic geometry, with significant implications for both theoretical understanding and practical computation.