Weakening the topology of a Lie group

With any topological group ( G , U ) (G, \mathcal {U}) one can associate a locally arcwise-connected group ( G , U ∗ ) (G, {\mathcal {U}}^{\ast }) , where U ∗ {\mathcal {U}}^{\ast } is stronger than U \mathcal {U} . ( G , U ) (G, \mathcal {U}) is a weakened Lie ( W L ) (WL) group if ( G , U ∗ ) (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 , J ) (L,\mathcal {J}) is associated are completely determined by a certain abelian subgroup H H of L L which is called decisive. If L L has closed adjoint image, then H H is the center Z ( L ) Z(L) of L L ; otherwise, H H is the product of a vector group V V and a group J J that contains Z ( L ) Z(L) . J / Z ( L ) J/Z(L) is finite (trivial if L L is solvable). We also discuss the connection between these theorems and recent results of Goto.