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In this paper, we classify the modules over $\\W_n^+$ and over $\\W_n$ which are free $U(\\h_n)$-modules of rank $1$. These are the $\\W_n^+$-modules $\\Omega(\\Lambda_{n},a, S) $ for some $\\Lambda_n=(\\lambda_1,\\cdots,\\lambda_n) \\in (\\C^*)^n, a\\in \\C$, and $S\\subset \\{1,2,..., n\\}$; and $\\W_n$-modules $\\Omega(\\Lambda_n,a)$ for some $\\Lambda_n\\in (\\C^*)^n$ and some $a\\in \\C.$"},"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":"1401.1120","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-01-06T15:44:33Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"89d9e10c52e4d0d7c482c0213359eab88acd422f1baa96c3aa341b3509c67e5f","abstract_canon_sha256":"7677c4e5f9979e0c366f082f6b45b2188caf01bb365719ba17752b15b783769e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:08.075954Z","signature_b64":"DTzmockyfV3HaL5KV75MKMZq8uhckmbjCauilj9At0CxjXb/pyczQGL5r5FYdLEMyFMkGLMmbZ+Ss8KbeQC3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5f13ba1d63131e185f01d2db9d317699561cf860b4efd1349cacdb427dddf28","last_reissued_at":"2026-05-18T02:27:08.075148Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:08.075148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\W_n^+$- and $W_n$-module structures on $U(h)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RT","authors_text":"Haijun Tan, Kaiming Zhao","submitted_at":"2014-01-06T15:44:33Z","abstract_excerpt":"Let $\\h_n$ be the Cartan subalgebra of the Witt algebras $\\W_n^+=\\text{Der}\\C[t_1, t_2, ..., t_n]$ and $\\W_n=\\text{Der}\\C[t_1^{\\pm 1},t_2^{\\pm 1},\\cdots,t_n^{\\pm1}]$ where $1\\le n\\le \\infty$. In this paper, we classify the modules over $\\W_n^+$ and over $\\W_n$ which are free $U(\\h_n)$-modules of rank $1$. These are the $\\W_n^+$-modules $\\Omega(\\Lambda_{n},a, S) $ for some $\\Lambda_n=(\\lambda_1,\\cdots,\\lambda_n) \\in (\\C^*)^n, a\\in \\C$, and $S\\subset \\{1,2,..., n\\}$; and $\\W_n$-modules $\\Omega(\\Lambda_n,a)$ for some $\\Lambda_n\\in (\\C^*)^n$ and some $a\\in \\C.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1120","kind":"arxiv","version":2},"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":"1401.1120","created_at":"2026-05-18T02:27:08.075277+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.1120v2","created_at":"2026-05-18T02:27:08.075277+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.1120","created_at":"2026-05-18T02:27:08.075277+00:00"},{"alias_kind":"pith_short_12","alias_value":"UXYTXIOWGEY6","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"UXYTXIOWGEY6DBPQ","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"UXYTXIOW","created_at":"2026-05-18T12:28:52.271510+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/UXYTXIOWGEY6DBPQDUW3TUYXNG","json":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG.json","graph_json":"https://pith.science/api/pith-number/UXYTXIOWGEY6DBPQDUW3TUYXNG/graph.json","events_json":"https://pith.science/api/pith-number/UXYTXIOWGEY6DBPQDUW3TUYXNG/events.json","paper":"https://pith.science/paper/UXYTXIOW"},"agent_actions":{"view_html":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG","download_json":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG.json","view_paper":"https://pith.science/paper/UXYTXIOW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.1120&json=true","fetch_graph":"https://pith.science/api/pith-number/UXYTXIOWGEY6DBPQDUW3TUYXNG/graph.json","fetch_events":"https://pith.science/api/pith-number/UXYTXIOWGEY6DBPQDUW3TUYXNG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG/action/storage_attestation","attest_author":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG/action/author_attestation","sign_citation":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG/action/citation_signature","submit_replication":"https://pith.science/pith/UXYTXIOWGEY6DBPQDUW3TUYXNG/action/replication_record"}},"created_at":"2026-05-18T02:27:08.075277+00:00","updated_at":"2026-05-18T02:27:08.075277+00:00"}