A \(\lambda\)-infinitesimal bialgebra is both an algebra an a coalgebra, with the compatibility \[ \Delta(ab)=a\Delta(b)+\Delta(b)+\lambda (a\otimes b). \] For \(\lambda=0\), this gives infinitesimal bialgebras in the sense of S. A. Joni and G. C. Rota [Stud. Appl. Math. 61, 93–139 (1979; Zbl 0471.05020)] and M. Aguiar [Contemp. Math. 267, 1–29 (2000; Zbl 0982.16028)]; for \(\lambda=1\), this gives infinitesimal bialgebras in the sense of J.-L. Loday and M. O. Ronco [Adv. Math. 139, No. 2, 293–309 (1998; Zbl 0926.16032)].
Generalizing A. Connes and D. Kreimer’s construction on rooted trees [Commun. Math. Phys. 199, No. 1, 203–242 (1998; Zbl 0932.16038)] and the reviewer’s [Bull. Sci. Math. 126, No. 3, 193–239 (2002; Zbl 1013.16026); ibid. 126, No. 4, 249–288 (2002; Zbl 1013.16027)] and R. Holtkamp’s non commutative version [Arch. Math. 80, No. 4, 368–383 (2003; Zbl 1056.16030)], the algebra of decorated plane rooted trees is given a new coproduct \(\Delta_\epsilon\), making it a \(\lambda\)-infinitesimal bialgebra. For this coproduct, the grafting operators are 1-cocycles for the Cartier-Quillen cohomology (up to the weight \(\lambda\)), as in the Connes-Kreimer case. It is proved that these infinitesimal bialgebras are the free objects in the category of infinitesimal bialgebra given a family of 1-cocycles, generalizing the universal property of the non commutative Connes-Kreimer Hopf algebra.