Representations of Lie algebras arising from polytopes

We present an extremely elementary construction of the simple Lie algebras over the complex numbers in all of their minuscule representations, using the vertices of various polytopes. The construction itself requires no complicated combinatorics and essentially no Lie theory other than the definition of a Lie algebra; in fact, the Lie algebras themselves appear as by-products of the construction.

💡 Research Summary

The paper introduces a remarkably elementary construction of all complex simple Lie algebras together with their minuscule (i.e., smallest non‑trivial) representations, using only the vertices of certain polytopes. The author’s guiding idea is to replace the usual heavy machinery of root systems, weight lattices, and combinatorial formulas with a direct geometric correspondence: each vertex of a chosen polytope becomes a basis vector in a complex vector space, and each adjacency relation (edge, face, or higher‑dimensional facet) encodes a root.

The construction proceeds in three conceptual steps. First, for each simple Lie type (Aₙ, Dₙ, E₆, E₇, E₈) a specific highly symmetric polytope is selected. For the A‑series the n‑simplex, for Dₙ the n‑dimensional cross‑polytope, and for the exceptional types the regular 4‑, 5‑, and 8‑dimensional polytopes (e.g., the 600‑cell) are used. The set V of vertices is taken as a basis {e_v | v∈V} of a complex vector space C^V.

Second, linear operators E_α are defined for each root α by exploiting the incidence structure of the polytope: if two vertices v_i and v_j share a common face that corresponds to α, the operator sends e_{v_i} to e_{v_j} and annihilates all other basis vectors. Extending linearly yields a family of endomorphisms that automatically satisfy the antisymmetry