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On five dimensional Sasakian Lie algebras with trivial center. (English) Zbl 1391.53065

A Sasakian Lie algebra is a Lie algebra \({\mathfrak g}\) endowed with a quadruple \((\varphi ,\xi,\eta,g)\) where \(\varphi \in \mathrm{End}({\mathfrak g})\), \(\xi \in {\mathfrak g}\), \(\eta \in {\mathfrak g}^*\), and \(g\) is an inner product such that \[ \eta (\xi )=1,\;\varphi ^2 =-\mathrm{Id}+\eta \otimes \xi , \]
\[ g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y), \]
\[ g(X,\varphi Y)=d\eta (X,Y),\;N_\varphi =-2d\eta \otimes \xi, \] where \[ N\varphi (X,Y)=\varphi ^2 [X,Y]+[\varphi X,\varphi Y]-\varphi [\varphi X,Y]-\varphi [X,\varphi Y]. \] It is proven in this paper that a five-dimensional Sasakian Lie algebra \(\mathfrak g\) with trivial center is isomorphic to one of the following Lie algebra \[ \mathrm{sl} (2,{\mathbb R})\times \operatorname{aff} ({\mathbb R}),\;\mathrm{su}(2)\times \operatorname{aff}({\mathbb R}),\;{\mathbb R}^2\ltimes {\mathfrak h}_3, \] where \(\operatorname{aff}({\mathbb R})\) is the Lie algebra of the Lie group of affine motions of \({\mathbb R}\), and \({\mathfrak h}_3\) is the real three dimensional Heisenberg Lie algebra. Furthermore such a Lie algebra \(\mathfrak g\) is \(\varphi \)-symmetric. If \(G\) is the simply connected Lie group with Lie algebra \(\mathfrak g\), and \(H\) is the one-parameter subgroup generated by \(\xi \), then \(G/H\) is a Hermitian symmetric space.

MSC:

53C35 Differential geometry of symmetric spaces
22E60 Lie algebras of Lie groups
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References:

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