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In the set-up where R is a subalgebra of B, which is quite common, two nondegenerate pairings of S and T will define an anti-automorphism \\tau of the algebra S. Making use of a two-sided depth two structure, we prove that \\tau is an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra.\n  A final section discusses how \\tau and the nondegenerate pairings generalize to modules via the pi-method for depth two, and a certain derived mapping of c","authors_text":"Lars Kadison","cross_cats":["math.RA"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2007-01-15T22:09:34Z","title":"Skew Hopf algebras, irreducible extensions and the pi-method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701427","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f3bf1e2770cd7ad920cd18ae55e63ba058bf749d4d3f83096fa14aaceebaefe6","target":"record","created_at":"2026-05-18T04:08:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7b46f71f42c15dec7ad4eff81ef54be0d2831bbd197ffa2657f0f65be378135c","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2007-01-15T22:09:34Z","title_canon_sha256":"797563c3aabd170d2937b8b24f4d8ab1787c1026d79712d76a4853ccd18c30f0"},"schema_version":"1.0","source":{"id":"math/0701427","kind":"arxiv","version":2}},"canonical_sha256":"97511b459d8406aa514783c4fcbff31843bc6f7f4da4202440bffebc518cea77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97511b459d8406aa514783c4fcbff31843bc6f7f4da4202440bffebc518cea77","first_computed_at":"2026-05-18T04:08:52.963657Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:52.963657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2JfYc1f2/Wemvde0p+u8luZFpMC3D6oa0RSe+cXWDayRuB02Z3XS3vHs6aoJAwrXEKJKci+ZZyoso+YNaA9BAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:52.964263Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701427","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3bf1e2770cd7ad920cd18ae55e63ba058bf749d4d3f83096fa14aaceebaefe6","sha256:d331a4a3fb13d27b15fbff1847e1a58c523c781e902ab20ad8d3e45617659afb"],"state_sha256":"580baff2d315bd386a8232cb327a8f8e85499c302cdf59ef90bde8f2da409ec5"}