Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables

Let $z=(z_1,z_2,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ be noncommutative free variables and $t$ a formal parameter which commutes with $z$. Let $k$ be any unital integral domain of any characteristic and $F_t(z)=z-H_t(z)$ with $H_t(z)\in {k[[t]]< < z

}^{\times n}$andtheorder$o(H_t(z))\geq 2$\mathrm{.}Notethat$F_t(z)$ can be viewed as a deformation of the formal map $F(z):=z-H_{t=1}(z)$ when it makes sense (for example, when $H_t(z)\in {k[t]<<z>>}^{\times n}$). The inverse map $G_t(z)$ of $F_t(z)$ can always be written as $G_t(z)=z+M_t(z)$ with $M_t(z)\in {k[[t]]<<z>>}^{\times n}$ and $o(M_t(z))\ge 2$. In this paper, we first derive the PDE’s satisfied by $M_t(z)$ and $u(F_t), u(G_t)\in {k[[t]]< < z >>}$with$u(z)\in {k< < z >>}$ in the general case as well as in the special case when $H_t(z)=tH(z)$ for some $H(z)\in {k<<z>>}^{\times n}$. We also show that the formal power series 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.