{"paper":{"title":"Cocommutative Hopf Dialgebras and Rack Combinatorics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For every cocommutative Hopf dialgebra the set-like rack of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup of its group-like elements.","cross_cats":[],"primary_cat":"math.RA","authors_text":"Andr\\'es Sarrazola-Alzate, Jos\\'e Gregorio Rodr\\'iguez-Nieto, Olga Patricia Salazar-D\\'iaz, Ra\\'ul Vel\\'asquez","submitted_at":"2026-05-12T20:56:35Z","abstract_excerpt":"We study cocommutative Hopf dialgebras through generalized digroups and rack combinatorics. We prove that the rack functor obtained from the adjoint rack bialgebra factorizes through the digroup of group-like elements. More precisely, for every cocommutative Hopf dialgebra $A$, the rack of set-like elements of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup $\\Glike(A)$. For finite generalized digroups $D\\simeq G\\times E$, with $G$ acting on the halo $E$, we derive explicit formulas for the conjugation rack, its inner group, left-translation cycle index"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every cocommutative Hopf dialgebra A, the rack of set-like elements of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup Glike(A).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The factorization of the rack functor through the digroup of group-like elements relies on the cocommutativity assumption and on the existence of a well-defined adjoint rack bialgebra structure, both of which are taken as given without further justification in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For cocommutative Hopf dialgebras the set-like rack is naturally isomorphic to the conjugation rack of the group-like digroup, and every finite generalized digroup arises as the group-like elements of its digroup algebra.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For every cocommutative Hopf dialgebra the set-like rack of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup of its group-like elements.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e32452f6d60c8c5a40d9baf8d9c739c61f06d2f0b82bc97c0d691caecb52b835"},"source":{"id":"2605.12749","kind":"arxiv","version":1},"verdict":{"id":"98662fd3-9d3a-40e2-9153-d01759a4880f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:32:18.117722Z","strongest_claim":"For every cocommutative Hopf dialgebra A, the rack of set-like elements of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup Glike(A).","one_line_summary":"For cocommutative Hopf dialgebras the set-like rack is naturally isomorphic to the conjugation rack of the group-like digroup, and every finite generalized digroup arises as the group-like elements of its digroup algebra.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The factorization of the rack functor through the digroup of group-like elements relies on the cocommutativity assumption and on the existence of a well-defined adjoint rack bialgebra structure, both of which are taken as given without further justification in the abstract.","pith_extraction_headline":"For every cocommutative Hopf dialgebra the set-like rack of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup of its group-like elements."},"references":{"count":23,"sample":[{"doi":"10.4172/1736-4337.1000244","year":2016,"title":"C. Alexandre, M. Bordemann, S. Rivière and F. Wagemann, Structure theory of rack- bialgebras,Journal of Generalized Lie Theory and Applications10(2016), no. 1, Art. ID 1000244, 1–20. DOI: 10.4172/1736","work_id":"140814c6-7dc1-4345-a997-ee54fc9838c2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/s0001-8708(02)00071-3","year":2003,"title":"N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras,Advances in Mathe- matics178(2003), no. 2, 177–243. DOI: 10.1016/S0001-8708(02)00071-3","work_id":"d5d69439-b34f-48d6-9e2c-a0a3cf2b666e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/s0002-9947-03-03046-0","year":2003,"title":"J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-suminvariantsofknottedcurvesandsurfaces,Transactions of the American Mathematical Society355(2003), no. 10","work_id":"37e4488b-f228-4820-95b3-508b166f2ed0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/s0022-4049(02)00159-7","year":2003,"title":"P. Etingof and M. Graña, On rack cohomology,Journal of Pure and Applied Algebra177 (2003), no. 1, 49–59. DOI: 10.1016/S0022-4049(02)00159-7","work_id":"e02f9508-a94a-44f2-b4f7-e3bd9681f08a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/bf00872903","year":1995,"title":"R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces,Applied Categorical Structures3(1995), no. 4, 321–356. DOI: 10.1007/BF00872903","work_id":"2eb68624-ef70-41c5-90ee-f8ffba9727f6","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"49a30879a162208bdafa7dc9bbb31e8034d5fb412587c4d2faf84130470b44e2","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"}