{"paper":{"title":"Dimension Quotients of Metabelian Lie Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Inder Bir S. Passi, Thomas Sicking","submitted_at":"2016-02-16T06:04:59Z","abstract_excerpt":"For a Lie ring $L$ over the ring of integers, we compare its lower central series $\\{\\gamma_n(L)\\}_{n\\geq 1}$ and its dimension series $\\{\\delta_n(L)\\}_{n\\geq 1}$ defined by setting $\\delta_n(L)= L\\cap \\varpi^n(L)$, where $\\varpi(L)$ is the augmentation ideal of the universal enveloping algebra of $L$. While $\\gamma_n(L)\\subseteq\\delta_n(L)$ for all $n\\geq 1$, the two series can differ. In this paper it is proved that if $L$ is a metabelian Lie ring, then $2\\delta_n(L)\\subseteq\\gamma_n(L)$, and $[\\delta_n(L),\\,L]=\\gamma_{n+1}(L)$, for all $n\\geq 1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04919","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"}