A Lie group corresponding to the free Lie algebra and its universality
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916665469435904 |
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| author | Neretin, Yury A. |
| author_facet | Neretin, Yury A. |
| contents | Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $ω_1$, \dots, $ω_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff formula, we have a corresponding Lie group $\overline{\mathrm{Fr}}_n$. It is the set $\exp\bigl(\overline{\mathfrak{fr}}_n\bigr)$ in the completed universal enveloping algebra of $\mathfrak{fr}_n$. Also, the group $\overline{\mathrm{Fr}}_n$ is a 'submanifold' in the algebra of formal associative noncommutative series in $ω_1$, \dots, $ω_n$, the 'submanifold' is determined by a certain system of quadratic equations. We consider a certain dense subgroup $\mathrm{Fr}_n^\infty\subset \overline{\mathrm{Fr}}_n$ with a stronger (Polish) topology and show that any homomorphism $π$ from $\mathfrak{fr}_n$ to a real finite-dimensional Lie algebra $\mathfrak{g}$ can be integrated in a unique way to a homomorphism $Π$ from $\mathrm{Fr}_n^\infty$ to the corresponding simply connected Lie group $G$. If $π$ is surjective, then $Π$ also is surjective. Note that Pestov (1993) constructed a separable Banach--Lie group such that any separable Banach--Lie group is its quotient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Lie group corresponding to the free Lie algebra and its universality Neretin, Yury A. Group Theory Rings and Algebras Representation Theory 17B35, 22E15, 22E65, 17B01 Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $ω_1$, \dots, $ω_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff formula, we have a corresponding Lie group $\overline{\mathrm{Fr}}_n$. It is the set $\exp\bigl(\overline{\mathfrak{fr}}_n\bigr)$ in the completed universal enveloping algebra of $\mathfrak{fr}_n$. Also, the group $\overline{\mathrm{Fr}}_n$ is a 'submanifold' in the algebra of formal associative noncommutative series in $ω_1$, \dots, $ω_n$, the 'submanifold' is determined by a certain system of quadratic equations. We consider a certain dense subgroup $\mathrm{Fr}_n^\infty\subset \overline{\mathrm{Fr}}_n$ with a stronger (Polish) topology and show that any homomorphism $π$ from $\mathfrak{fr}_n$ to a real finite-dimensional Lie algebra $\mathfrak{g}$ can be integrated in a unique way to a homomorphism $Π$ from $\mathrm{Fr}_n^\infty$ to the corresponding simply connected Lie group $G$. If $π$ is surjective, then $Π$ also is surjective. Note that Pestov (1993) constructed a separable Banach--Lie group such that any separable Banach--Lie group is its quotient. |
| title | A Lie group corresponding to the free Lie algebra and its universality |
| topic | Group Theory Rings and Algebras Representation Theory 17B35, 22E15, 22E65, 17B01 |
| url | https://arxiv.org/abs/2411.11184 |