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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weakening the topology of a Lie group


Author: T. Christine Stevens
Journal: Trans. Amer. Math. Soc. 276 (1983), 541-549
MSC: Primary 22E20; Secondary 22A05, 22E15
MathSciNet review: 688961
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Abstract: With any topological group $ (G, \mathcal{U})$ one can associate a locally arcwise-connected group $ (G, {\mathcal{U}}^{\ast})$, where $ {\mathcal{U}}^{\ast}$ is stronger than $ \mathcal{U}$. $ (G, \mathcal{U})$ is a weakened Lie $ (WL)$ group if $ (G, {\mathcal{U}}^{\ast})$ is a Lie group. In this paper the author shows that the WL groups with which a given connected Lie group $ (L,\mathcal{J})$ is associated are completely determined by a certain abelian subgroup $ H$ of $ L$ which is called decisive. If $ L$ has closed adjoint image, then $ H$ is the center $ Z(L)$ of $ L$; otherwise, $ H$ is the product of a vector group $ V$ and a group $ J$ that contains $ Z(L)$. $ J/Z(L)$ is finite (trivial if $ L$ is solvable). We also discuss the connection between these theorems and recent results of Goto.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0688961-1
PII: S 0002-9947(1983)0688961-1
Keywords: Lie group, locally arcwise-connected group, (CA) analytic group
Article copyright: © Copyright 1983 American Mathematical Society