PreprintarXiv licencearXiv
Three Generations in E7
Abstract
Starting from the Standard Model Lie algebra gSM = sl3 ⊕ sl2 ⊕ ℂ sitting inside the complex Lie algebra e7, we show how to decompose e7 into the direct sum of a Lie subalgebra containing gSM and three 32-dimensional subspaces, each of which forms the same representation of gSM as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the E7 generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding gSM ⊂ e7, and also the description of each 32-dimensional subspace as a copy of the exterior algebra Λℂ5.
Related
- The three Kanade-Russell identities modulo nine
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- 144 real circles tangent to three conics
- A short proof of the complete Crouzeix inequality in dimension three
BibTeX
@misc{baez2026three,
title = {{Three Generations in E7}},
author = {John C. Baez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06271},
eprint = {2608.06271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06271},
}
RIS
TY - UNPB
TI - Three Generations in E7
AU - John C. Baez
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.06271
UR - https://arxiv.org/abs/2608.06271
AB - Starting from the Standard Model Lie algebra g_SM = sl_3 ⊕ sl_2 ⊕ ℂ sitting inside the complex Lie algebra e_7, we show how to decompose e_7 into the direct sum of a Lie subalgebra containing g_SM and three 32-dimensional subspaces, each of which forms the same representation of g_SM as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the E_7 generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding g_SM ⊂ e_7, and also the description of each 32-dimensional subspace as a copy of the exterior algebra Λℂ^5.
ER -