Integral Structures on H-type Lie Algebras

TL;DR

This paper proves that all H-type Lie algebras have an integral basis, leading to the existence of cocompact lattices in their simply connected groups and enabling explicit lattice constructions and isoperimetric dimension calculations.

Contribution

It establishes the existence of integral bases for H-type Lie algebras and explicitly constructs lattices, simplifying the understanding of their geometric and algebraic properties.

Findings

Existence of integral bases for all H-type Lie algebras

Explicit construction of cocompact lattices in H-type groups

Calculation of isoperimetric dimensions for H-type groups

Abstract

In this paper we prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal'cev criterion that all simply connected H-type Lie groups contain cocompact lattices. Since the Campbell-Hausdorff formula is very simple for two-step nilpotent Lie groups we can actually avoid invoking the Mal'cev criterion and exhibit our lattices in an explicit way. As an application, we calculate the isoperimetric dimensions of H-type groups.