{"paper":{"title":"$\\mathbb{Z}_2^3$-grading of the Lie algebra $G_2$ and related color algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.MP","math.RA"],"primary_cat":"math-ph","authors_text":"J. Van der Jeugt, N.I. Stoilova","submitted_at":"2025-05-07T12:59:32Z","abstract_excerpt":"We present a special and attractive basis for the exceptional Lie algebra $G_2$, which turns $G_2$ into a $\\mathbb{Z}_2^3$-graded Lie algebra. There are two basis elements for each degree of $\\mathbb{Z}_2^3\\setminus\\{(0,0,0)\\}$, thus yielding 14 basis elements. We give a general and simple closed form expression for commutators between these basis elements. Next, we use this $\\mathbb{Z}_2^3$-grading in order to examine graded color algebras. Our analysis yields three different $\\mathbb{Z}_2^3$-graded color algebras of type $G_2$. Since the $\\mathbb{Z}_2^3$-grading is not compatible with a Cart"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.04378","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/2505.04378/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"}