Deformations and Inversion Formulas for Formal Automorphisms in Noncommutative Variables

Let z = (z1, z2, · · · , zn) be noncommutative free variables and t a formal parameter which commutes with z. Let k be a unital commutative ring of any characteristic and Ft(z) = z−Ht(z) with Ht(z) ∈ k[[t]]〈〈z〉〉 ×n and o(Ht(z)) ≥ 2. Note that Ft(z) can be viewed as a deformation of the formal map F (z) := z−Ht=1(z) when it makes sense. The inverse map Gt(z) of Ft(z) can always be written as Gt(z) = z + Mt(z) with Mt(z) ∈ k[[t]]〈〈z〉〉 ×n and o(Mt(z)) ≥ 2. In this paper, we first derive the PDE’s satisfied by Mt(z) and u(Ft), u(Gt) ∈ k[[t]]〈〈z〉〉 with u(z) ∈ k〈〈z〉〉 in the general case as well as in the special case when Ht(z) = tH(z) for some H(z) ∈ k〈〈z〉〉×n. We also show that the elements above are actually characterized by certain Cauchy problems of these PDE’s. Secondly, We apply the derived PDE’s to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char.k = 0, we derive an expansion inversion formula by the planar binary rooted trees.