Geodesic
Algebraic Structures on Typed Decorated Rooted Trees
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes–Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes–Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned–Hairer–Zambotti construction as a subquotient.
Keywords: typed tree, combinatorial Hopf algebras, operads.
Mots-clés : pre-Lie algebras 
@article{SIGMA_2021_17_a85,
     author = {Lo{\"\i}c Foissy},
     title = {Algebraic {Structures} on {Typed} {Decorated} {Rooted} {Trees}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a85/}
}
TY  - JOUR
AU  - Loïc Foissy
TI  - Algebraic Structures on Typed Decorated Rooted Trees
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2021
VL  - 17
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a85/
LA  - en
ID  - SIGMA_2021_17_a85
ER  - 
%0 Journal Article
%A Loïc Foissy
%T Algebraic Structures on Typed Decorated Rooted Trees
%J Symmetry, integrability and geometry: methods and applications
%D 2021
%V 17
%U http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a85/
%G en
%F SIGMA_2021_17_a85
Loïc Foissy. Algebraic Structures on Typed Decorated Rooted Trees. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a85/

[1] Bergeron F., Labelle G., Leroux P., Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998 | DOI

[2] Bremner M. R., Dotsenko V., Algebraic operads. An algorithmic companion, CRC Press, Boca Raton, FL, 2016

[3] Bruned Y., Singular KPZ type equations, https://tel.archives-ouvertes.fr/tel-01306427

[4] Bruned Y., Chandra A., Chevyrev I., Hairer M., “Renormalising SPDEs in regularity structures”, J. Eur. Math. Soc. (JEMS), 23 (2021), 869–947, arXiv: 1711.10239 | DOI

[5] Bruned Y., Hairer M., Zambotti L., “Algebraic renormalisation of regularity structures”, Invent. Math., 215 (2019), 1039–1156, arXiv: 1610.08468 | DOI

[6] Bruned Y., Manchon D., Algebraic deformation for SPDEs, arXiv: 2011.05907

[7] Calaque D., Ebrahimi-Fard K., Manchon D., “Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series”, Adv. in Appl. Math., 47 (2011), 282–308, arXiv: 0806.2238 | DOI

[8] Chapoton F., “Un endofoncteur de la catégorie des opérades”, Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 105–110 | DOI

[9] Chapoton F., Livernet M., “Pre-Lie algebras and the rooted trees operad”, Int. Math. Res. Not., 2001 (2001), 395–408, arXiv: math.QA/0002069 | DOI

[10] Connes A., Kreimer D., “Hopf algebras, renormalization and noncommutative geometry”, Comm. Math. Phys., 199 (1998), 203–242, arXiv: hep-th/9808042 | DOI

[11] Dotsenko V., Griffin J., “Cacti and filtered distributive laws”, Algebr. Geom. Topol., 14 (2014), 3185–3225, arXiv: 1109.5345 | DOI

[12] Foissy L., Algebraic structures associated to operads, arXiv: 1702.05344

[13] Foissy L., Generalized prelie and permutative algebras, arXiv: 2104.00909

[14] Grossman R., Larson R. G., “Hopf-algebraic structure of combinatorial objects and differential operators”, Israel J. Math., 72 (1990), 109–117 | DOI

[15] Hoffman M. E., “Combinatorics of rooted trees and Hopf algebras”, Trans. Amer. Math. Soc., 355 (2003), 3795–3811, arXiv: math.CO/0201253 | DOI

[16] Livernet M., “A rigidity theorem for pre-Lie algebras”, J. Pure Appl. Algebra, 207 (2006), 1–18, arXiv: math.QA/0504296 | DOI

[17] Loday J.-L., Vallette B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, Heidelberg, 2012 | DOI

[18] Manchon D., Zhang Y., Free pre-Lie family algebras, arXiv: 2003.00917

[19] Markl M., Shnider S., Stasheff J., Operads in algebra, topology and physics, Mathematical Surveys and Monographs, 96, Amer. Math. Soc., Providence, RI, 2002 | DOI

[20] Mathar R. J., Topologically distinct sets of non-intersecting circles in the plane, arXiv: 1603.00077

[21] Oudom J.-M., Guin D., “Sur l'algèbre enveloppante d'une algèbre pré-Lie”, C. R. Math. Acad. Sci. Paris, 340 (2005), 331–336 | DOI

[22] Oudom J.-M., Guin D., “On the Lie enveloping algebra of a pre-Lie algebra”, J. K-Theory, 2 (2008), 147–167, arXiv: math.QA/0404457 | DOI

[23] Panaite F., “Relating the Connes–Kreimer and Grossman–Larson Hopf algebras built on rooted trees”, Lett. Math. Phys., 51 (2000), 211–219, arXiv: math.QA/0003074 | DOI

[24] Sloane N. J.A., The on-line encyclopedia of integer sequences, http://oeis.org/

[25] Zhang Y., Gao X., Guo L., “Matching Rota–Baxter algebras, matching dendriform algebras and matching pre-Lie algebras”, J. Algebra, 552 (2020), 134–170, arXiv: 1909.10577 | DOI