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Non-associative Hopf Algebra on Dyck Paths Arising from the Shuffle-insertion Product — arXiv:2610.38139

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.38139
License (as archived): CC0-1.0
Archived: 2026-11-01T02:37:38.024Z
SHA-256 of the archived text: 5a82ca96a04c5033ed874e497bb0a6e935ea6a7ecc55a4a2978002afb113da16
Stable URL of this page: https://metraipolis.com/documento/dc_25998f5bf655

How to cite

Non-associative Hopf Algebra on Dyck Paths Arising from the Shuffle-insertion Product — arXiv:2610.38139 — arXiv (en), https://arxiv.org/abs/2610.38139 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_25998f5bf655 (Metraipolis archive dc_25998f5bf655, sha256 5a82ca96a04c5033ed874e497bb0a6e935ea6a7ecc55a4a2978002afb113da16)

BibTeX

@misc{dc25998f5bf655,
  title = {Non-associative Hopf Algebra on Dyck Paths Arising from the Shuffle-insertion Product — arXiv:2610.38139},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.38139},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_25998f5bf655; sha256 5a82ca96a04c5033ed874e497bb0a6e935ea6a7ecc55a4a2978002afb113da16},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.38139. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this paper, we define a shuffle-insertion product and a cut-box coproduct on the vector space spanned by Dyck paths, where the product is commutative but non-associative. We prove that the product and coproduct are compatible, making the space a non-associative bialgebra. This bialgebra is connected and bigraded by path degree and peak number, and hence is a non-associative Hopf algebra. Its primitive subspace is spanned by atoms, and associators of primitive elements are primitive. Furthermore, we provide closed-form formulas for its left and right antipodes. We also define the convolution product on the space of linear transformations of the non-associative Hopf algebra, which equips this space with a non-associative algebra structure. Finally, we construct a one-parameter deformation preserving the cut-box coproduct. Its zero specialization is the ordinary shuffle Hopf algebra, which is also the associated graded algebra for the atom filtration, while every nonzero specialization is isomorphic to the shuffle-insertion algebra.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.38139 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:15:28.243Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).