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This isomorphism is shown to hold whenever $e_1$ is subregular, and in $\\mathfrak{sl}_n$ for all $n \\le 4$.\n  We next define embeddings of various categories $\\mathcal{O}$ for the W-algebras associated to $e_1$ and $e_2$, amongst them the embeddings $\\mathcal{O}(e_2,\\mathfrak{p}) \\hookrightarrow \\mathcal{O}("},"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":"1510.07352","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-10-26T02:29:18Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"3b555f479d2621a5b0bda29442c12881551ae3295df68c98249d4ac76ce44e78","abstract_canon_sha256":"8a3a5d6ae293e81a6c873f2e86d9e7108dbe875825f2e37667897454472d6caf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:14.288026Z","signature_b64":"NhqwsGj/G/PFhK2c1lj9OQx9S96SodT+JXHXSpsICFJcV3F1WQVf5HzP6V1glSP8FKrX2SzTHVTWSlckzLecBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cf602556edab575619e3b756ae5fc04408442b1a93a34bfea5334d371157202","last_reissued_at":"2026-05-18T01:29:14.287553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:14.287553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantum Hamiltonian reduction of W-algebras and category O","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RT","authors_text":"Stephen Morgan","submitted_at":"2015-10-26T02:29:18Z","abstract_excerpt":"We define a quantum version of Hamiltonian reduction by stages, producing a construction in type A for a quantum Hamiltonian reduction from the W-algebra $U(\\mathfrak{g},e_1)$ to an algebra conjecturally isomorphic to $U(\\mathfrak{g},e_2)$, whenever $e_2 \\ge e_1$ in the dominance ordering. 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