Three Generations in E7

Abstract

Starting from the Standard Model Lie algebra ${\mathfrak{g}}_{\mathrm{SM}}={\mathfrak{sl}}_3\oplus {\mathfrak{sl}}_2\oplus \mathbb{C}$ sitting inside the complex Lie algebra ${\mathfrak{e}}_7$, we show how to decompose ${\mathfrak{e}}_7$ into the direct sum of a Lie subalgebra containing ${\mathfrak{g}}_{\mathrm{SM}}$ and three 32-dimensional subspaces, each of which forms the same representation of ${\mathfrak{g}}_{\mathrm{SM}}$ as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the ${\mathrm{E}}_7$ generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding ${\mathfrak{g}}_{\mathrm{SM}}\subset {\mathfrak{e}}_7$, and also the description of each 32-dimensional subspace as a copy of the exterior algebra $\Lambda {\mathbb{C}}^5$.

17 pages LaTeX with TikZ figures