## Osaka Journal of Mathematics

### Invariant complex structures on tangent and cotangent Lie groups of dimension six

#### Abstract

This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type $\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\mathrm{T}_{\pi} \mathfrak{h}$ for $\mathfrak{h}$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constraint $J\mathfrak{h} = V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non-singular derivations of $\mathfrak{h}$. This fact allows different kinds of applications.

#### Article information

Source
Osaka J. Math., Volume 49, Number 2 (2012), 489-513.

Dates
First available in Project Euclid: 20 June 2012
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