Cofree Com-PreLie algebras

Year 2025, Volume: 38 Issue: 38, 139 - 183, 14.07.2025

Loïc Foissy

https://doi.org/10.24330/ieja.1608563

Abstract

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the preLie product is homogeneous of degree $\geq -1$. We also give a graphical description of free unitary Com-PreLie algebras, explicit their canonical bialgebra structure and exhibit with the help of a rigidity theorem certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted trees. We finally prove that the dual of these bialgebras are also enveloping algebras of preLie algebras, combinatorially described.

Keywords

PreLie algebra , cofree bialgebra , Connes-Kreimer Hopf algebra , Connes-Moscovici Hopf algebra

References

• T. Benes and D. Burde, Degenerations of pre-Lie algebras, J. Math. Phys., 50(11) (2009), 112102 (9 pp).
• D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, Comm. Math. Phys., 215(1) (2000), 217-236.
• A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199(1) (1998), 203-242.
• L. Foissy, Finite-dimensional comodules over the Hopf algebra of rooted trees, J. Algebra, 255(1) (2002), 89-120.
• L. Foissy, The Hopf algebra of Fliess operators and its dual pre-Lie algebra, Comm. Algebra, 43(10) (2015), 4528-4552.
• L. Foissy, A pre-Lie algebra associated to a linear endomorphism and related algebraic structures, Eur. J. Math., 1(1) (2015), 78-121.
• W. S. Gray and L. A. Duffaut Espinosa, A Faa di Bruno Hopf algebra for a group of Fliess operators with applications to feedback}, Systems Control Lett., 60(7) (2011), 441-449.
• M. Livernet, A rigidity theorem for pre-Lie algebras, J. Pure Appl. Algebra, 207(1) (2006), 1-18.
• J.-L. Loday, Splitting associativity and Hopf algebras, Actes des journees mathematiques a la memoire de Jean Leray, Soc. Math. France, Paris, Semin. Congr., 9 (2004), 155-172.
• J.-L. Loday and M. Ronco, On the structure of cofree Hopf algebras, J. Reine Angew. Math., 592 (2006), 123-155.
• J.-L. Loday and M. Ronco, Combinatorial Hopf algebras, Quanta of maths, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 11 (2010), 347-383.
• J.-M. Oudom and D. Guin, Sur l'algebre enveloppante d'une algebre pre-Lie, C. R. Math. Acad. Sci. Paris, 340(5) (2005), 331-336.
• J.-M. Oudom and D. Guin, On the Lie enveloping algebra of a pre-Lie algebra, J. \(K\)-Theory., 2(1) (2008), 147-167.
• N. J. A. Sloane, The on-line encyclopedia of integer sequences, \url{https://oeis.org/}.