Pith Number

pith:SL2M25KE

pith:2026:SL2M25KEN5G5JIAIAKWYHAPQIY

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Cocommutative Hopf Dialgebras and Rack Combinatorics

Andr\'es Sarrazola-Alzate, Jos\'e Gregorio Rodr\'iguez-Nieto, Olga Patricia Salazar-D\'iaz, Ra\'ul Vel\'asquez

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.

arxiv:2605.12749 v1 · 2026-05-12 · math.RA

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Claims

C1strongest 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).

C2weakest 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.

C3one 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.

References

23 extracted · 23 resolved · 0 Pith anchors

[1] 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 2016 · doi:10.4172/1736-4337.1000244

[2] 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 2003 · doi:10.1016/s0001-8708(02)00071-3

[3] 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 2003 · doi:10.1090/s0002-9947-03-03046-0

[4] 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 2003 · doi:10.1016/s0022-4049(02)00159-7

[5] R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces,Applied Categorical Structures3(1995), no. 4, 321–356. DOI: 10.1007/BF00872903 1995 · doi:10.1007/bf00872903

Receipt and verification

First computed	2026-05-18T03:09:48.880334Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

92f4cd75446f4dd4a00802ad8381f0463a6f958f5b9070dba400d87476120b55

Aliases

arxiv: 2605.12749 · arxiv_version: 2605.12749v1 · doi: 10.48550/arxiv.2605.12749 · pith_short_12: SL2M25KEN5G5 · pith_short_16: SL2M25KEN5G5JIAI · pith_short_8: SL2M25KE

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/SL2M25KEN5G5JIAIAKWYHAPQIY \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 92f4cd75446f4dd4a00802ad8381f0463a6f958f5b9070dba400d87476120b55

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f74285b7e0e640f27d268da8d9a1744dfb4b52343a2ffa8e466c86b2a8c34a28",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.RA",
    "submitted_at": "2026-05-12T20:56:35Z",
    "title_canon_sha256": "a6865ee0570006f450819a3dc24de414991249c24077e1ef7f84a24e8541f7ba"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12749",
    "kind": "arxiv",
    "version": 1
  }
}