{"paper":{"title":"Three Generations in E7","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"John C. Baez","submitted_at":"2026-08-06T17:01:41Z","abstract_excerpt":"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 o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06271","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06271/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}