## 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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