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A special left Alia algebra (resp. coalgebra) $(A, [,]_{(f,g)})$ (resp. $(A, \\Delta_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A, \\cdot)$ (resp. $(A, \\delta)$) together with two linear maps $f, g: A\\longrightarrow A$ (resp. $F, G: A\\longrightarrow A$). We find that if $((A, \\cdot), f)$ (resp. $((A, \\delta), F)$) is a Nijenhuis associative "},"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":"2506.07061","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2025-06-08T09:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"f8a497641d073d66a0dd7e796455b34769552c53b6bfe6804224e44a1d597b04","abstract_canon_sha256":"d1e70301455d8ae5e5719afc8a7c4eb3062838147d8fd6e52389e3642b585e74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:00.629006Z","signature_b64":"XV21lWOWEghaOtWUqisMXqpvMuyy7AfLS3LO9eo7CpA0d2bDMuL0kF1BJbYH2TCXJVqTxciEvJGpJYSVSD/WCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83726b22077bf65131d9185c1f9722b72c21ea342eed7607af367d74558d3c90","last_reissued_at":"2026-07-05T11:18:00.628496Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:00.628496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bialgebras induced by special left Alia algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Chan Zhao, Huihui Zheng, Tianshui Ma","submitted_at":"2025-06-08T09:44:25Z","abstract_excerpt":"Special left Alia algebras were introduced by Dzhumadil'daev in [J. Math. Sci. (N.Y.) 161(2009), 11-30] when studying the classification of algebras with skew-symmetric identity of degree 3. A special left Alia algebra (resp. coalgebra) $(A, [,]_{(f,g)})$ (resp. $(A, \\Delta_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A, \\cdot)$ (resp. $(A, \\delta)$) together with two linear maps $f, g: A\\longrightarrow A$ (resp. $F, G: A\\longrightarrow A$). 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