Complete classification of pseudo H-type Lie algebras: I

Abstract

Let ${\mathscr {N}}$ be a 2-step nilpotent Lie algebra endowed with a 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. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra ${{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})$, and the representation map $J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)$ 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 depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters $0\le r,s<\infty$. We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.