Journal of Lie Theory

no. 3
On Epimorphisms in some Categories of Infinite-Dimensional Lie Groups
Journal of Lie Theory, Volume 31 (2021) no. 3, pp. 871-884
Let $X$ be a smooth compact connected manifold. Let $G=\text{Diff}\,X$ be the group of diffeomorphisms of $X$, equipped with the $C^\infty$-topology, and let $H$ be the stabilizer of some point in $X$. Then the inclusion $H\to G$, which is a morphism of two regular Fr\'echet-Lie groups, is an epimorphism in the category of smooth Lie groups modelled on complete locally convex spaces. At the same time, in the latter category, epimorphisms between finite dimensional Lie groups have dense range. We also prove that if $G$ is a Banach-Lie group and $H$ is a proper closed subgroup, the inclusion $H\to G$ is not an epimorphism in the category of Hausdorff topological groups.
DOI: 10.5802/jolt.1198
Classification: 18A20, 22E65, 58D05
Keywords: Epimorphism, locally convex Lie group, Frechet-Lie group, Banach-Lie group, Hausdorff topological group 
@article{JOLT_2021_31_3_a10,
     author = {V. G. Pestov and V. V. Uspenskij},
     title = {On {Epimorphisms} in some {Categories} of {Infinite-Dimensional} {Lie} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {871--884},
     year = {2021},
     volume = {31},
     number = {3},
     doi = {10.5802/jolt.1198},
     zbl = {1486.22016},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1198/}
}
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%A V. V. Uspenskij
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V. G. Pestov; V. V. Uspenskij. On Epimorphisms in some Categories of Infinite-Dimensional Lie Groups. Journal of Lie Theory, Volume 31 (2021) no. 3, pp. 871-884. doi: 10.5802/jolt.1198

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