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On closedness and simple connectedness of adjoint and coadjoint orbits. (English) Zbl 0815.22004
Let $$\mathfrak g$$ be a finite dimensional real Lie algebra and $$\mathfrak h$$ a subalgebra of $$\mathfrak g$$. For a subset $$S$$ of a group we write $$\langle S\rangle$$ for the subgroup generated by $$S$$. We define $$\text{Inn}_{\mathfrak g}({\mathfrak h}) := \langle e^{\text{ad}{\mathfrak h}}\rangle$$ and $$\text{INN}_{\mathfrak g}({\mathfrak h}) := \overline{\text{Inn}_{\mathfrak g}({\mathfrak h})}$$. We also set $$\text{Inn}_{\mathfrak g} := \text{Inn}_{\mathfrak g}({\mathfrak g})$$ and $$\text{INN}_{\mathfrak g} := \text{INN}_{\mathfrak g}({\mathfrak g})$$. Then the adjoint action is the action of $$\text{Inn}_{\mathfrak g}$$ on the Lie algebra $$\mathfrak g$$ and the coadjoint action is the action of $$\text{Inn}_{\mathfrak g}$$ on $${\mathfrak g}^*$$ via $$g.v := (g^{-1})^*(v) = v \circ g^{-1}$$. The main results of this paper are: (1) Every coadjoint orbit of strict convexity type is closed and simply connected; (2) Every adjoint orbit meeting a Cartan subalgebra $$\mathfrak t$$ and every coadjoint orbit meeting $${\mathfrak t}^*$$ is closed and one component of its Zariski closure.
Reviewer: A.K.Guts (Omsk)

##### MSC:
 22E60 Lie algebras of Lie groups 17B05 Structure theory for Lie algebras and superalgebras
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