Invariant integral structures in pseudo $H$-type Lie algebras: construction and classification

TL;DR

This paper develops a classification method for integral orthonormal structures in pseudo H-type Lie algebras, leading to a full classification for certain parameters and implications for discrete subgroups.

Contribution

It introduces a new classification approach for integral structures in pseudo H-type Lie algebras and applies it to a broad parameter range, extending via Atiyah-Bott periodicity.

Findings

Complete classification for r=1 to 16, s=0 or 1, with irreducible Clifford modules.

Existence of integral structures implies discrete uniform subgroups.

Framework facilitates further extensions using periodicity.

Abstract

Pseudo $H$-type Lie algebras are a special class of 2-step nilpotent metric Lie algebras, intimately related to Clifford algebras $\Cl_{r,s}$. In this work we propose the classification method for integral orthonormal structures of pseudo $H$-type Lie algebras. We apply this method for the full classification of these structures for $r\in\{1,\ldots,16\}$, $s\in \{0,1\}$ and irreducible Clifford modules. The latter cases form the basis for the further extensions by making use of the Atiyah-Bott periodicity. The existence of integral structures gives rise to the integral discrete uniform subgroups of the pseudo $H$-type Lie groups.