Complete classification of $H$-type algebras: I

Let $\mathscr N$ be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle.\,,.\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the centre of the Lie algebra and $V$ 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 $\Cl(\mathbb R^{r,s})$ and the representation map $J\colon \Cl(\mathbb R^{r,s})\to(V)$ is related to the Lie algebra structure by $\langle J_zv,w\rangle=\langle z,[v,w]\rangle$ 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 $0\leq r,s<\infty$. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.

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