Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 066, 36 pages      arXiv:2207.03839      https://doi.org/10.3842/SIGMA.2023.066

Tridendriform Structures

Pierre Catoire
Université du Littoral Côte d'Opale, UR 2597 LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62100 Calais, France

Received November 10, 2022, in final form August 31, 2023; Published online September 15, 2023

Abstract
Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and we show that they have a coproduct satisfying some compatibilities with the tridendriform products. Its graded dual is the opposite bialgebra of TSym introduced by N. Bergeron et al., which is described by the lightening splitting of a tree. In particular, we can split the product in three pieces and the coproduct in two pieces with Hopf compatibilities. We generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work.

Key words: Hopf algebras; tridendriform; dendriform; Schröder trees.

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References

  1. Aguiar M., Bergeron N., Sottile F., Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), 1-30, arXiv:math.CO/0310016.
  2. Aguiar M., Sottile F., Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra 295 (2006), 473-511, arXiv:math.CO/0409022.
  3. Bergeron N., González D'León R.S., Li S.X., Pang C.Y.A., Vargas Y., Hopf algebras of parking functions and decorated planar trees, Adv. in Appl. Math. 143 (2023), 102436, 62 pages, arXiv:2109.05434.
  4. Burgunder E., Delcroix-Oger B., Structure theorems for dendriform and tridendriform algebras, in Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA). Vol. 1, IRMA Lect. Math. Theor. Phys., Vol. 31, EMS Publ. House, Berlin, 2020, 29-66.
  5. Burgunder E., Ronco M., Tridendriform structure on combinatorial Hopf algebras, J. Algebra 324 (2010), 2860-2883, arXiv:0910.0403.
  6. Chapoton F., Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces, J. Pure Appl. Algebra 168 (2002), 1-18, arXiv:math.QA/0005253.
  7. Ebrahimi-Fard K., Patras F., Monotone, free, and boolean cumulants: A shuffle algebra approach, Adv. Math. 328 (2018), 112-132, arXiv:1701.06152.
  8. Foissy L., Algèbres de Hopf combinatoires, http://loic.foissy.free.fr/pageperso/Hopf.pdf.
  9. Foissy L., Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra 209 (2007), 439-459, arXiv:math.RA/0505207.
  10. Foissy L., Free brace algebras are free pre-Lie algebras, Comm. Algebra 38 (2010), 3358-3369, arXiv:0809.2866.
  11. Foissy L., Patras F., Lie theory for quasi-shuffle bialgebras, in Periods in Quantum Field Theory and Arithmetic, Springer Proc. Math. Stat., Vol. 314, Springer, Cham, 2020, 483-540, arXiv:1605.02444.
  12. Loday J.-L., Scindement d'associativité et algèbres de Hopf, in Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr., Vol. 9, Société Mathématique de France, Paris, 2004, 155-172, arXiv:math.QA/0402084.
  13. Loday J.-L., On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), 79-93, arXiv:math.QA/0506498.
  14. Loday J.-L., Ronco M., Algèbres de Hopf colibres, C. R. Math. Acad. Sci. Paris 337 (2003), 153-158.
  15. Loday J.-L., Ronco M., Trialgebras and families of polytopes, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory, Contemp. Math., Vol. 346, American Mathematical Society, Providence, RI, 2004, 369-398, arXiv:math.AT/0205043.
  16. Loday J.-L., Ronco M.O., Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293-309.
  17. Novelli J.-C., Thibon J.-Y., Binary shuffle bases for quasi-symmetric functions, Ramanujan J. 40 (2016), 207-225, arXiv:1305.5032.
  18. Palacios P., Ronco M.O., Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra 299 (2006), 648-678.
  19. Ronco M., Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra 254 (2002), 152-172.
  20. Sloane N.J.A., The on-line encyclopedia of integers sequences, https://oeis.org/.
  21. Zhang Y., Gao X., Manchon D., Free (tri)dendriform family algebras, J. Algebra 547 (2020), 456-493, arXiv:1909.08946.

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