On complex H-type Lie algebras

Abstract

Let $\mathfrak{g}$ be a complex nilpotent Lie algebra equipped with a
Hermitian inner product $\langle \cdot, \cdot\rangle$.  We show that
if $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$ is an H-type Lie
algebra in the sense of Kaplan, then $\mathfrak{g}$ must be isomorphic
to a complex Heisenberg Lie algebra
$\mathfrak{h}^{2n+1}_{\mathbb{C}}$.  This shows that the class of
complex H-type Lie algebras is very small.