Complete classification of $H$-type algebras: I

Let $\mathcal{N}$ be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product $⟨\mathrm{.},\mathrm{.}⟩$ and let $\mathscr N=V\oplus_{\perp}Z$,where$Z$isthecentreoftheLiealgebraand$V$ its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space $V$ arises as a representation space of a Clifford algebra $\text{Cl}({\mathbb{R}}^{r,s})$ and the representation map $J\text{\hspace{0.17em}⁣}:\text{Cl}({\mathbb{R}}^{r,s})\to (V)$ is related to the Lie algebra structure by $⟨J_{z}v,w⟩=⟨z,[v,w]⟩$ for all $z\in {\mathbb{R}}^{r,s}$ and $v,w\in V$. The classification is based on the range of parameters $r$ and $s$ and is completed for the Clifford modules $V$, having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters $0\le r,s<\mathrm{\infty }$. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.