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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 $n\\geq 1$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.04919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-02-16T06:04:59Z","cross_cats_sorted":[],"title_canon_sha256":"6129a618cbb26f67da465396c0b280372b1287d08ebb76a51d0e135645b59ae1","abstract_canon_sha256":"f3d235dc742ce52d3a929d49b51be753694360ccad317cfceefec162e32df28a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:37.860394Z","signature_b64":"3FaIzLwiwFHUsMi+/PDTfOlaKbXNv82kcjGzLwrSk5zBAdENN7Gh8M+/QDMKhKpKNv8zucvHCM0mjT/35gJ4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0679b73583e70c39ce1e9f74ec1ff416ae5dd67f9dd34099e33dc9aed0e6d06a","last_reissued_at":"2026-05-18T01:20:37.859982Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:37.859982Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.04919","created_at":"2026-05-18T01:20:37.860047+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.04919v1","created_at":"2026-05-18T01:20:37.860047+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04919","created_at":"2026-05-18T01:20:37.860047+00:00"},{"alias_kind":"pith_short_12","alias_value":"AZ43ONMD44GD","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"AZ43ONMD44GDTTQ6","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"AZ43ONMD","created_at":"2026-05-18T12:30:07.202191+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2","json":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2.json","graph_json":"https://pith.science/api/pith-number/AZ43ONMD44GDTTQ6T52OYH7UC2/graph.json","events_json":"https://pith.science/api/pith-number/AZ43ONMD44GDTTQ6T52OYH7UC2/events.json","paper":"https://pith.science/paper/AZ43ONMD"},"agent_actions":{"view_html":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2","download_json":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2.json","view_paper":"https://pith.science/paper/AZ43ONMD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.04919&json=true","fetch_graph":"https://pith.science/api/pith-number/AZ43ONMD44GDTTQ6T52OYH7UC2/graph.json","fetch_events":"https://pith.science/api/pith-number/AZ43ONMD44GDTTQ6T52OYH7UC2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2/action/storage_attestation","attest_author":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2/action/author_attestation","sign_citation":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2/action/citation_signature","submit_replication":"https://pith.science/pith/AZ43ONMD44GDTTQ6T52OYH7UC2/action/replication_record"}},"created_at":"2026-05-18T01:20:37.860047+00:00","updated_at":"2026-05-18T01:20:37.860047+00:00"}