Complete classification of pseudo $H$-type algebras: II

Abstract

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $J\colon \Cl(\mathbb R^{r,s})\toU$ be a representation of the Clifford algebra $\Cl(\mathbb R^{r,s})$ generated by the pseudo Euclidean vector space $\mathbb R^{r,s}$. Assume that the Clifford module $U$ is endowed with a bilinear symmetric non-degenerate real form $\la\cdot\,,\cdot\ra_U$ making the linear map $J_z$ skew symmetric for any $z\in\mathbb R^{r,s}$. The Lie algebras and the Clifford algebras are related by $\la J_zv,w\ra_U=\la z,[v,w]\ra_{\mathbb R^{r,s}}$, $z\in \mathbb R^{r,s}$, $v,w\in U$. We detect the isomorphic and non-isomorphic Lie algebras according to the dimension of $U$ and the range of the non-negative integers~$r,s$.