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Let $k(G)$, respectively, $k(\\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\\mathfrak{g})/k(\\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\\mathfrak{g})/k(\\mathfrak{g})^G$, a"},"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":"0901.4358","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-01-27T21:48:22Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"872594342e0bae310482d7626068ad01c8a68d34a9f5d12320c5554408d65ee6","abstract_canon_sha256":"dbd094ead67ede0047d218363e5f65e326d8fa7958ed6440dd9bdf3c8b0cf0b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:44.776702Z","signature_b64":"0pelZyTj0usJLNEOvV5BfGllyq1M4ZG2UOwc/scMUYJJOPYe+y4I6AwctgVNj9/1vddRnw/WaDfODusZFiwmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84f5d0e890fae3fe22a5bcd276ec942f5c54ff7cf5ad4ae1a537934a3d6386e3","last_reissued_at":"2026-05-18T03:02:44.775975Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:44.775975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"Boris Kunyavski\\u{i}, Jean-Louis Colliot-Th\\'el\\`ene, Vladimir L. Popov, Zinovy Reichstein","submitted_at":"2009-01-27T21:48:22Z","abstract_excerpt":"Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\\mathfrak{g})/k(\\mathfrak{g})^G$ are purely transcendental. 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