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Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology "},"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":"1710.11022","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-10-30T15:48:54Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"ef734f4d33f57a1d0408e6dcadbe4003a67260bb44e19e7235d9ae9e0082b454","abstract_canon_sha256":"59723bd9e27146a2432865ed7e6780e6c21d5003df3b15775a7fae327aef54af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:21.805135Z","signature_b64":"2kVHsw/M7P/T5x2sgi7Sv/FR/9Y61OyymZgQu264Cl0JNvY6CiUsYpFhq9aEGy2pJhTY5xOTIyEx4y9Tj9L+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af858dfcb4f9c71ee98af3c5374f9d131d510fb2539b5818c9c90da0fc7dbe3f","last_reissued_at":"2026-05-18T00:13:21.804619Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:21.804619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Rudolf Tange","submitted_at":"2017-10-30T15:48:54Z","abstract_excerpt":"Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G_1 and B_1. Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. 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