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We construct a 2-fold monoidal structure on the category of tetramodules of a bialgebra.\n  Suppose $C$ is an abelian $n$-fold monoidal category with the unit object $A$. We prove, provided some condition (*), that $Ext_C(A,A)$ is an $(n+1)$-algebra. In the case of bialgebras this condition (*) is satisfied when $A$ is a Hopf algebra. Finally, the GS cohomology of a Hopf algebra is a 3-algebra.\n  As well, we consider this kind of questions"},"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":"0907.3335","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-07-20T04:18:56Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"1cec3324ec53ff7550596b60a632079a0fb9188b21c9c24f780889a56400f498","abstract_canon_sha256":"dc8e13c769fce81c90f656faf798c159551f64c00563407de29dca45e28ed7fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:22:03.174433Z","signature_b64":"b0GCZlkMzfWxocvECyCDw5yZWg8rz+Xy+wcrhg4Q7dxlsh/6TkxVynPXwriFx6ii5c8g03MEfTxs27vqgtdMAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e60fdcd550f2bd42dff088b37adfdd181cfcf7d58b0f2a1be5d64619eb5da62","last_reissued_at":"2026-07-04T17:22:03.174021Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:22:03.174021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf algebras, tetramodules, and n-fold monoidal categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CT","authors_text":"Boris Shoikhet","submitted_at":"2009-07-20T04:18:56Z","abstract_excerpt":"The abelian category of tetramodules over an associative bialgebra $A$ is related with the Gerstenhaber-Schack (GS) cohomology as $Ext_\\Tetra(A,A)=H_\\GS(A)$. 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