Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables

Details Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables

2005-09-06

arxiv.org/abs/math/0509130v2

Internat. J. Algebra Comput., 17 (2007), no. 2, 261--288.

Mathematics

General Mathematics

Latex, 30 pages. Following the referee's suggestion, an example has been added and fully discussed. Some references have been

Scientific paper

Let $z=(z_1, z_2, ..., z_n)$ 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 $H_t(z)\in {k[[t]]< < z >>}^{\times n}$ and the order $o(H_t(z))\geq 2$. Note that $F_t(z)$ 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 $o(M_t(z))\geq 2$. 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.

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