Cocommutative Hopf Dialgebras and Rack Combinatorics
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| Format: | Preprint |
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2026
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| _version_ | 1866911677394452480 |
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| author | Rodríguez-Nieto, José Gregorio Salazar-Díaz, Olga Patricia Sarrazola-Alzate, Andrés Velásquez, Raúl |
| author_facet | Rodríguez-Nieto, José Gregorio Salazar-Díaz, Olga Patricia Sarrazola-Alzate, Andrés Velásquez, Raúl |
| contents | 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, fixed-point polynomial, orbit count and subrack structure. Finally, we construct the digroup algebra $K[D]$, prove that it is a cocommutative Hopf dialgebra, and show that $\Glike(K[D])=D\$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12749 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cocommutative Hopf Dialgebras and Rack Combinatorics Rodríguez-Nieto, José Gregorio Salazar-Díaz, Olga Patricia Sarrazola-Alzate, Andrés Velásquez, Raúl Rings and Algebras 16T05, 20N99, 20M10, 57K12 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, fixed-point polynomial, orbit count and subrack structure. Finally, we construct the digroup algebra $K[D]$, prove that it is a cocommutative Hopf dialgebra, and show that $\Glike(K[D])=D\$. |
| title | Cocommutative Hopf Dialgebras and Rack Combinatorics |
| topic | Rings and Algebras 16T05, 20N99, 20M10, 57K12 |
| url | https://arxiv.org/abs/2605.12749 |