The dual of infinitesimal unitary Hopf algebras and planar rooted forests

Yıl 2023, Cilt: 33 Sayı: 33, 178 - 204, 09.01.2023

Xiaomeng Wang Loic Foıssy Xing Gao

https://doi.org/10.24330/ieja.1220707

Öz

We study the infinitesimal (in the sense of Joni and Rota) bialgebra $H_{RT}$ of planar rooted trees introduced in a previous work
of two of the authors,
whose coproduct is given
by deletion of a vertex. We prove that its dual $H_{RT}^*$ is isomorphic to a free non unitary algebra, and give two free generating sets.
Giving $H_{RT}$ a second product, we make it an infinitesimal bialgebra in the sense of Loday and Ronco,
which allows to explicitly construct a projector onto its space of primitive elements, which freely generates $H_{RT}$.

Anahtar Kelimeler

Infinitesimal bialgebra, planar rooted trees, graftings, 1-cocycle

Kaynakça

• M. Aguiar, Infinitesimal Hopf algebras, Contemp. Math., 267 (2000), 1-29.
• Ch. Brouder, Trees, renormalization and differential equations, BIT, 44(3) (2004), 425-438.
• A. Connes and D. Kreimer, Hopf algebras, renormalization and non-commutative geometry, Comm. Math. Phys., 199(1) (1998), 203-242.
• R. Diestel, Graph theory, 5th Edition, (English) Graduate Texts in Mathematics 173, Berlin, Springer, 2017.
• L. Foissy, Les alg`ebres de Hopf des arbres enracin´es d´ecor´es I, Bull. Sci. Math., 126 (2002), 193-239.
• L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra, 209(2) (2007), 439-459.
• L. Foissy, The infinitesimal Hopf algebra and the poset of planar forests, J. Algebraic Combin., 30(3) (2009), 277-309.
• L. Foissy, Introduction to Hopf Algebra of Rooted Trees, available at http://loic. foissy. free. fr/pageperso/preprint3. pdf.
• X. Gao and X. M. Wang, Infinitesimal unitary Hopf algebras and planar rooted forests, J. Algebraic Combin., 49(4) (2019), 437-460.
• R. Grossman and R. G. Larson, Hopf-algebraic structure of families of trees, J. Algebra, 126(1) (1989), 184-210.
• L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, 4. International Press, 2012.
• M. Hoffman, Combinatorics of rooted trees and Hopf algebras, Trans. Amer. Math. Soc., 355(9) (2003), 3795-3811.
• R. Holtkamp, Comparison of Hopf algebras on trees, Arch. Math. (Basel), 80(4) (2003), 368-383.
• S. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61(2) (1979), 93-139.
• J.-L. Loday and M. O. Ronco, On the structure of cofree Hopf algebras, J. Reine Angew. Math., 592 (2006), 123-155.
• F. Panaite, Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees, Lett. Math. Phys., 51(3) (2000), 211-219.
• T. J. Zhang, X. Gao and L. Guo, Hopf algebras of rooted forests, cocyles and free Rota-Baxter algebras, J. Math. Phys., 57(10) (2016), 101701 (16 pp).