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Here $V_m$ is the irreducible representation of $\\mathfrak{sl}_2(\\Bbb C)$ of dimension $m+1$. For the computation of the cohomology we use the Hochschild-Serre formula, a long exact sequence in the cohomology and explicit formulas for the multiplicities of $V_k$ in the exterior product $\\Lambda^j(V_m)$ for $j\\le 4$. In general we cannot determ"},"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":"2411.14952","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-11-22T14:12:29Z","cross_cats_sorted":[],"title_canon_sha256":"42053035f3d1cc00841551259ba18c527382ca38b4a959146f9bd47167385cd9","abstract_canon_sha256":"3c07cc49800c62e19791aa73b0abd3a8e024a5984e8d0371c8032eef0db47476"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:39:14.095653Z","signature_b64":"itJGFW8H+kIyh/O3AYUksps4mTxhD/rTyzqi1kq7G3icAKZWGCDrWNSNlXn8DravAZwc/K4DMs9690cZnGOkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"076c0b14356bc678d0f5290fdf6a2f83513731dbe9b9b39b49acb0541b8250e2","last_reissued_at":"2026-07-05T09:39:14.095245Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:39:14.095245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cohomology of perfect Lie algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Dietrich Burde, Friedrich Wagemann","submitted_at":"2024-11-22T14:12:29Z","abstract_excerpt":"We study the adjoint cohomology of perfect Lie algebras over the complex numbers. 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