Ncs Systerms over Differential Operator Algebras and the Grossman-larson Hopf Algebras of Labeled Rooted Trees

Let K be any unital commutative Q-algebra and W any non-empty subset of N. Let z = (z1, . . . , zn) be commutative or noncommutative free variables and t a formal central parameter. Let D〈〈z〉〉 (α ≥ 1) be the unital algebra generated by the differential operators ofK〈〈z〉〉 which increase the degree in z by at least α− 1 and A [α] t 〈〈z〉〉 the group of automorphisms Ft(z) = z−Ht(z) of K[[t]]〈〈z〉〉 with o(Ht(z)) ≥ α and Ht=0(z) = 0. First, we study a connection of the NCS systems ΩFt (Ft ∈ A [α] t 〈〈z〉〉) ([Z5], [Z6]) over the differential operators algebra D〈〈z〉〉 and the NCS system Ω T ([Z8]) over the Grossman-Larson Hopf algebra H GL ([GL], [F1], [F2]) of W -labeled rooted trees. We construct a Hopf algebra homomorphism AFt : H W GL → D 〈〈z〉〉 (Ft ∈ A [α] t 〈〈z〉〉) such that A Ft (Ω W T ) = ΩFt . Secondly, we generalize the tree expansion formulas for the inverse map ([BCW], [Wr]), the D-Log and the formal flow ([WZ]) of Ft in the commutative case to the noncommutative case. Thirdly, we prove the injectivity of the specialization T : NSym → H + GL ([Z8]) of NCSF’s (noncommutative symmetric functions) ([GKLLRT]). Finally, we show the family of the specializations SFt of NCSF’s with all n ≥ 1 and the polynomial automorphisms Ft = z −Ht(z) with Ht(z) homogeneous and the Jacobian matrix JHt strictly lower triangular can distinguish any two different NCSF’s. The graded dualized versions of the main results above are also discussed. 2000 Mathematics Subject Classification. Primary: 05E05, 14R10, 16W30; Secondary: 16W20, 06A11.