Weakening the topology of a Lie group

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.