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Exponentiality of certain real solvable Lie groups. (English) Zbl 0913.22007

A connected Lie group \(G\) is called exponential if its exponential function \(\exp : {\mathbf g} \to G\) is surjective. In general it is quite hard to prove or disprove that a given Lie group \(G\) is exponential. In [Math. Nachr. 192, 255-267 (1998; Zbl 0904.22004)] the second author describes various characterizations of solvable exponential Lie groups. One condition characterizing exponential solvable Lie groups is that the centralizers of all \(ad\)-nilpotent elements in \({\mathbf g}\) in a Cartan subgroup \(H \subseteq G\) are connected. The first main result of the paper under review simplifies this criterion for groups of the type \(G = N \mathrel\times T\), where \(N\) is nilpotent and \(T\) a torus. It is shown that \(G\) is exponential if and only if the centralizer in \(T\) of each element in the Lie algebra of \(N\) is connected. The second main result is a more technical statement dealing with semidirect products \(N \mathrel\times T\), where \(N\) is solvable.

MSC:

22E25 Nilpotent and solvable Lie groups
22E15 General properties and structure of real Lie groups

Citations:

Zbl 0904.22004
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