Invariant complex structures on tangent and cotangent Lie groups of dimension six (Q436058)

Tangent and cotangent Lie algebras are semidirect products \(\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V\), where \(\pi\) is the adjoint or the co-adjoint representation, respectively. The present paper deals with left invariant complex structures on simply connected Lie groups, whose Lie algebra is a tangent or cotangent Lie algebra. As a start, the existence of totally real complex structures is studied, i.e., complex structures \(J\) satisfying the constraint \(J\mathfrak{h} = V\). It is shown that, if \(\pi\) is the adjoint representation, this kind of complex structures are associated to non-singular derivations of \(\mathfrak{h}\). Thus, the existence of totally real complex structures on \(\mathrm{T}_{\pi} \mathfrak{h}\) implies that \(\mathfrak{h}\) must be nilpotent. Then, the three dimensional case is studied in detail. For the coadjoint representation, the general form of totally real complex structures is given. Moreover, pseudo-Kähler structures are examined.