{"paper":{"title":"Skew Hopf algebras, irreducible extensions and the pi-method","license":"","headline":"","cross_cats":["math.RA"],"primary_cat":"math.QA","authors_text":"Lars Kadison","submitted_at":"2007-01-15T22:09:34Z","abstract_excerpt":"To a depth two extension A | B, we associate the dual bialgebroids S := \\End {}_BA_B and T := (A \\o_B A)^B over the centralizer R=C_A(B). 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701427","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}