Some remarks on the group of formal diffeomorphisms of the line
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908216052416512 |
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| author | Neretin, Yury A. |
| author_facet | Neretin, Yury A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some remarks on the group of formal diffeomorphisms of the line Neretin, Yury A. Representation Theory Functional Analysis Group Theory 22E65, 22E66, 17B35, 17B65, 17B68 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. |
| title | Some remarks on the group of formal diffeomorphisms of the line |
| topic | Representation Theory Functional Analysis Group Theory 22E65, 22E66, 17B35, 17B65, 17B68 |
| url | https://arxiv.org/abs/2502.05582 |