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In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}\\mathcal{YD}.$ Let $H_{R}$ be the Majid's transmuted braided group of $\\left( H,R\\right) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}\\oplus\\cdots\\oplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $\\left( 1\\leq i\\leq r\\right) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define th"},"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":"1811.05593","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-11-14T01:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"ff9a864827c0d9d6d078db3b2dbdb940d28803b35c6322607a6e72439005634a","abstract_canon_sha256":"06ab247a8af2f9a11944c4f2be836239aa224d288b14cf03b33efb46b915a255"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:17.230129Z","signature_b64":"q89DNrUfZlQGq47tfznBVBcCVBHyKiEf6hdiTORf/J0r1PmOgujkBaekW1gtcYECjQ6OFOrhI2VKDaT8+v4JBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2a7911b9a01d12ba4474ffd5eb5343406ea99dab2244384191f515484ede9f4","last_reissued_at":"2026-05-17T23:43:17.229404Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:17.229404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Structure of Irreducible Yetter-Drinfeld Modules over Quasi-Triangular Hopf Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Shenglin Zhu, Zhimin Liu","submitted_at":"2018-11-14T01:42:14Z","abstract_excerpt":"Let $\\left( H,R\\right) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. 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