Complex structures on affine motion groups

Abstract

Abstract. We study existence of complex structures on semidirect products g ⊕ ρ v where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl(2, C). We also exhibit a complex structure on the Galilean algebra G(3, 1). We construct next a complex structure on g ⊕ ρ v starting with one on g under certain compatibility assumptions on ρ.As an application of our results we obtain that there exists k ∈ {0, 1} such that (S 1 ) k × E(n) admits a left invariant complex structure, where S 1 is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group P 4k+3 has a natural left invariant complex structure.In case dim g = dim v, then there is an adapted complex structure on g ⊕ ρ v precisely when ρ determines a flat, torsion-free connection on g. If ρ is self-dual, g ⊕ ρ v carries a natural symplectic structure as well. If, moreover, ρ comes from a metric connection then g ⊕ ρ v possesses a pseudo-Kähler structure.We prove that the tangent bundle T G of a Lie group G carrying a flat torsion free connection ∇ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

From almost (para)-complex structures to affine structures on Lie groups

“…• They generalize previous results concerning complex product structures [2,7], complex and symplectic structures related to tangent algebras [1,6,17], complex and paracomplex structures on homogeneous manifolds [11]. • The existence of LSA structures imposes a clear obstruction.…”

Special Hermitian metrics and Lie groups

“…Since D is flat, the Lie bracket (2) on T D g satisfies the Jacobi identity. The integrability of the complex structureJ on T D g follows from the fact that J is integrable and parallel with respect to D (see [2,Proposition 3.3]).…”

“…To construct the tangent Lie algebra we consider the flat connection D as a representation of g on R 2n . If we choose the adjoint representation ad of g on g then the semidirect product g ⋉ ad R 2n is the Lie algebra of the Lie group T G, the tangent bundle over G [2]. In this case the conditions DJ = Dg = 0 are satisfied if and only if (g, J) is a complex Lie algebra and the inner product g is bi-invariant.…”

Lie bialgebras of complex type and associated Poisson Lie groups

“…When the bialgebra structure on g is trivial, that is, δ = 0, the Lie algebra Dg is the cotangent Lie algebra T * g. It was proved in [4] that if J is a complex structure on g and we define J as in (7), then J is a complex structure on the cotangent algebra T * g. Moreover, (J, · , · ) is a Hermitian structure on T * g, where · , · is the canonical ad-invariant neutral metric.…”

“…The first assertion is a consequence of Proposition 12. For (2), use the fact that J defined as in (7) is a complex structure on T * g (see [4]) and therefore (J, · , · ) is a Hermitian structure on T * g. The second assertion now follows by applying (1).…”