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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the dual of an exponential solvable Lie group

Author: Bradley N. Currey
Journal: Trans. Amer. Math. Soc. 309 (1988), 295-307
MSC: Primary 22E27
MathSciNet review: 957072
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Abstract: Let $ G$ be a connected, simply connected exponential solvable Lie group with Lie algebra $ \mathfrak{g}$. The Kirillov mapping $ \eta :\,\,\mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G) \to \hat G$ gives a natural parametrization of $ \hat G$ by co-adjoint orbits and is known to be continuous. In this paper a finite partition of $ \mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G)$ is defined by means of an explicit construction which gives the partition a natural total ordering, such that the minimal element is open and dense. Given $ \pi \in \hat G$, elements in the enveloping algebra of $ {\mathfrak{g}_c}$ are constructed whose images under $ \pi $ are scalar and give crucial information about the associated orbit. This information is then used to show that the restriction of $ \eta $ to each element of the above-mentioned partition is a homeomorphism.

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Article copyright: © Copyright 1988 American Mathematical Society