Integral Shell Polytopes of Composition Algebras

TL;DR

This paper explores the geometric structures of integral shells in composition algebras, revealing connections to root systems, polytopes, and the E8 lattice, with novel insights into Okubo algebra and lattice decompositions.

Contribution

It introduces the first detailed study of shell polytopes in real composition algebras, linking integral norms to classical and exceptional polytopes, and describes their lattice and symmetry properties.

Findings

Root-polytopal configurations linked to Hurwitz systems are recovered.

Okubo algebra exhibits a distinct hierarchy involving cross-polytopes and D8 root polytopes.

The E8 Gosset polytope is reconstructed via lattice gluing and symmetry analysis.

Abstract

Integral systems in real composition algebras give rise to finite metric configurations whose geometry is linked to both regular polytopes and root-systems. In this work we investigate, to our knowledge for the first time in this form, the shell polytopes obtained by fixing the integral norm and taking the convex hull of the corresponding integral elements. The first shells recover the familiar root-polytopal configurations attached to the classical Hurwitz systems, while the Okubo algebra gives a quite different behaviour. The Okubo integral closure does not recover the Gosset polytope directly: it selects a two-adic hierarchy whose first visible layers are a cross-polytope and a \(D_8\) root polytope. We further show that the natural intermediate lattice is isometric to the rescaled cubic lattice; consequently every shell decomposes into explicit orbits of the hyperoctahedral group \(W(B_8)\), and the higher Okubo shells admit a complete combinatorial description in cubic-lattice coordinates. The full \(E_8\) Gosset polytope is then recovered from the intermediate lattice by maximal-isotropic gluing along \((\ZZ/2)^4\). This gives an interplay between non-unital composition, integral lattice shadows, and the geometry of \(E_8\).