发表机构
University of Edinburgh; University of California, Riverside（爱丁堡大学; 加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于E₇李代数，将其分解为含标准模型李代数的子代数与三个对应费米子代的32维子空间，推导仅依赖标准模型李代数在E₇中的嵌入，为E₇代统一理论提供新基础。
AI 中文摘要
从复李代数𝔢₇内的标准模型李代数𝔤_SM = 𝔰𝔩₃ ⊕ 𝔰𝔩₂ ⊕ ℂ出发，本文展示如何将𝔢₇分解为一个包含𝔤_SM的李子代数与三个32维子空间的直和，每个子空间作为𝔤_SM的相同表示，对应一代费米子及其反粒子。该设定源于Nasmith，所用数学大多是Kugo与Yanagida的E₇代统一理论的基础。新特点在于推导尽可能仅依赖嵌入𝔤_SM ⊂ 𝔢₇，且将每个32维子空间描述为外代数Λℂ⁵的一个副本。
英文摘要
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 $Λ\mathbb{C}^5$.
Comments17 pages LaTeX with TikZ figures
