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Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables | | Wenhua Zhao
; | | Date: |
6 Sep 2005 | | Subject: | General Mathematics; Complex Variables MSC-class: 14R10, 32H02 | math.GM math.CV | | Abstract: | Let $z=(z_1, z_2, ..., z_n)$ be noncommutative free variables and $t$ a formal parameter which commutes with $z$. Let $k$ be a unital commutative ring of any characteristic and $F_t(z)=z-H_t(z)$ with $H_t(z)in kttzz^{ imes n}$ and $o(H_t(z))geq 2$. 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. The inverse map $G_t(z)$ of $F_t(z)$ can always be written as $G_t(z)=z+M_t(z)$ with $M_t(z)in kttzz^{ imes n}$ and $o(M_t(z))geq 2$. In this paper, we first derive the PDE’s satisfied by $M_t(z)$ and $u(F_t), u(G_t)in kttzz$ with $u(z)in kzz$ in the general case as well as in the special case when $H_t(z)=tH(z)$ for some $H(z)in kzz^{ imes n}$. 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. | | Source: | arXiv, math.GM/0509130 | | Services: | Forum | Review | PDF | Favorites | |
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