Free nilpotent and $H$-type Lie algebras. Combinatorial and orthogonal designs

TL;DR

This paper constructs pseudo H-type Lie algebras from free nilpotent two-step Lie algebras using combinatorial methods, providing explicit algorithms and establishing their rational structures and lattice existence.

Contribution

It introduces an explicit algorithm for constructing pseudo H-type algebras via ideals, linking algebraic structures with combinatorial and orthogonal designs.

Findings

Established a method to construct pseudo H-type algebras from free nilpotent Lie algebras.

Proved the existence of rational structures and lattices on pseudo H-type Lie groups.

Connected pseudo H-type algebras with combinatorial and orthogonal design frameworks.

Abstract

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices.