{"paper":{"title":"Cartan's Magic Formula for Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CA","authors_text":"Oliver Knill","submitted_at":"2018-11-26T00:27:21Z","abstract_excerpt":"Cartan's magic formula L_X = i_X d + d i_X = (d+i_X)^2=D_X^2 relates the exterior derivative d, an interior derivative i_X and its Lie derivative L_X. We use this formula to define a finite dimensional vector space of vector fields X on a finite abstract simplicial complex G. This space has a Lie algebra structure satisfying L_[X,Y] = L_X L_Y - L_Y L_X as in the continuum. Any such vector field X defines a coordinate change on the finite dimensional vector space l^2(G) which play the role of translations along the vector field. If i_X^2=0, the relation L_X=D_X^2 with D_X=i_X+d mirrors the Hodg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10125","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":""},"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"}