Existence of the lattice on general $H$-type groups

Abstract

Let $\mathscr N$ be a two step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle\cdot\,,\cdot\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the center of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We prove that if $(V,\langle\cdot\,,\cdot\rangle_V)$ is the Clifford module for the Clifford algebra $\Cl(Z,\langle\cdot\,,\cdot\rangle_Z)$ such that the homomorphism $J\colon \Cl(Z,\langle\cdot\,,\cdot\rangle_Z)\to\End(V)$ is skew symmetric with respect to the scalar product $\langle\cdot\,,\cdot\rangle_V$, or in other words the Lie algebra $\mathscr N$ satisfies conditions of general $H$-type Lie algebras ~\cite{Ciatti, GKM}, then there is a basis with respect to which the structural constants of the Lie algebra $\mathscr N$ are all $\pm 1$ or 0.