Some remarks on the group of formal diffeomorphisms of the line
Yury A. Neretin
TL;DR
The paper addresses lifting representations of a strictly positively graded Lie algebra $\mathfrak g$ to pro-unipotent completions and applies this to the group $\overline{\mathbf{Diff}}$ of formal diffeomorphisms of the line. It constructs a Polish-topology dense subgroup $\mathbf{G}^\circ$ that admits liftings to finite-dimensional and to bounded Banach-space representations, and specializes to $\overline{\mathbf{Diff}}$ to identify $\mathbf{Diff}^\circ$ as the group of series with subfactorial coefficient growth. It provides analytic structures on diffeomorphism groups, including an exponential map from a Banach-Lie algebra $\boldsymbol{\mathfrak{vect}}^\circ$ to $\mathbf{Diff}^\circ$ and a local logarithm, and situates $\mathbf{Diff}^\circ$ within Ner1's completion framework, showing universality properties. The work connects formal-diffeomorphism topology with Rashevsky-type completions and offers a concrete, representation-theoretic pathway to integrate bounded representations of $\mathfrak vect$ to $\mathbf{Diff}^\circ$, with potential applications to monodromy and related geometric structures.
Abstract
Consider a strictly positively graded finitely generated infinite-dimensional real Lie algebra $\mathfrak{g}$. It has a well-defined Lie group $\overline{\mathbf{G}}$, which is an inverse limit of finite-dimensional nilpotent Lie groups (a pro-unipotent group). Generally, representations (even finite-dimensional representations) of $\mathfrak{g}$ and actions of $\mathfrak{g}$ on manifolds do not admit liftings to $\overline{\mathbf{G}}$. There is a canonically defined dense subgroup $\mathbf{G}^\circ\subset \overline{\mathbf{G}}$ with a stronger (Polish) topology, which admits lifting of representations of $\mathfrak{g}$ in finite-dimensional spaces (and, more generally, of representations of $\mathfrak{g}$ by bounded operators in Banach spaces). We describe this completion for the group $\overline{\mathbf{Diff}}$ of formal diffeomorphisms of the line, i.e., substitutions of the form $x\mapsto x+ p(x)$, where $p(x)=a_2 x^2+\dots$ are formal series, and show that the group $\mathbf{Diff}^\circ$ consists of series with subfactorial growth of coefficients.