Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Magic Star of Exceptional Periodicity

Piero Truini, Michael Rios, Alessio Marrani

TL;DR

Exceptional Periodicity (EP) introduces a periodic, finite-dimensional generalisation of the exceptional structures, forming a chain parameterised by $n$ that extends $\mathbf{e_8}$, $\mathbf{J_3^{\mathbb{O}}}$, and $\mathbb{O}$. The framework uses extended root systems and the Magic Star projection to define finite algebras ${\mathcal{L}_{MS}}$ at each level and Vinberg's cubic $T$-algebras to realise a non-Lie, Jordan-like structure, with $n=1$ recovering the classical MS. It realises $\mathbf{J_3^{\mathbb{O}}}$ as a member of a family of $T$-algebras and places $\mathbf{e_8}$ at the core of a Magic Star of extended algebras, suggesting links to lattice vertex algebras and noncommutative geometry. The authors outline potential pathways to quantum gravity through emergent spacetime, modular forms, and higher-dimensional lattice constructions, proposing a broad algebraic toolkit that extends exceptional symmetries beyond traditional Lie theory.

Abstract

We present a periodic infinite chain of finite generalisations of the exceptional structures, including e8, the exceptional Jordan algebra (and pair), and the octonions. We demonstrate that the exceptional Jordan algebra is part of an infinite family of finite-dimensional matrix algebras (corresponding to a particular class of cubic Vinberg's T-algebras). Correspondingly, we prove that e8 is part of an infinite family of algebras (dubbed "Magic Star" algebras) that resemble lattice vertex algebras.

The Magic Star of Exceptional Periodicity

TL;DR

Exceptional Periodicity (EP) introduces a periodic, finite-dimensional generalisation of the exceptional structures, forming a chain parameterised by that extends , , and . The framework uses extended root systems and the Magic Star projection to define finite algebras at each level and Vinberg's cubic -algebras to realise a non-Lie, Jordan-like structure, with recovering the classical MS. It realises as a member of a family of -algebras and places at the core of a Magic Star of extended algebras, suggesting links to lattice vertex algebras and noncommutative geometry. The authors outline potential pathways to quantum gravity through emergent spacetime, modular forms, and higher-dimensional lattice constructions, proposing a broad algebraic toolkit that extends exceptional symmetries beyond traditional Lie theory.

Abstract

We present a periodic infinite chain of finite generalisations of the exceptional structures, including e8, the exceptional Jordan algebra (and pair), and the octonions. We demonstrate that the exceptional Jordan algebra is part of an infinite family of finite-dimensional matrix algebras (corresponding to a particular class of cubic Vinberg's T-algebras). Correspondingly, we prove that e8 is part of an infinite family of algebras (dubbed "Magic Star" algebras) that resemble lattice vertex algebras.

Paper Structure

This paper contains 7 sections, 7 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Jordan Structures and related Lie algebras
  3. The Magic Star: Exceptional Lie algebras and Jordan structures
  4. Exceptional Periodicity and Extended Roots
  5. The ${\bf{\mathcal{L}_{MS}}}$ algebra
  6. T-algebras
  7. Future Developments

Key Result

Proposition 4.2

For all $\rho\in \Phi_O$ and $x\in \Phi$: $2\dfrac{(x, \rho)}{(\rho,\rho)}\in \mathbb Z$ and $w_\rho(x) = x - 2\dfrac{(x, \rho)}{(\rho,\rho)}\rho\in \Phi$ (the set of extended roots is closed under the Weyl reflections by all $\rho\in \Phi_O$). The set of extended roots is closed under the Weyl refl

Figures (2)

  • Figure 1: Root diagram of $\mathbf{f_4}$, $\mathbf{e_6}$, $\mathbf{e_7}$, $\mathbf{e_8}$ (${\pmb \nu}=1,2,4,8$) projected on the plane of one $\mathbf{a_2}$
  • Figure 4: The Magic Star

Theorems & Definitions (16)

  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • Definition 5.1
  • Proposition 5.2
  • proof
  • ...and 6 more