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

Abstract

Let $\mathcal{N}$ be a two step nilpotent Lie algebra endowed with non-degenerate scalar product $⟨\cdot ,\cdot ⟩$ and let $\mathcal{N}=V{\oplus }_{\perp }Z$, where $Z$ is the center of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We prove that if $(V,⟨\cdot ,\cdot {⟩}_V)$ 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 $⟨\cdot ,\cdot {⟩}_V$, or in other words the Lie algebra $\mathcal{N}$ 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 $\mathcal{N}$ are all $\pm 1$ or 0.

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Cited by in corpus (3)

• Complete classification of $H$-type algebras: I
• Structure constants of pseudo $H$-type algebras in some integral bases
• Classification of pseudo $H$-type algebras