A right-invariant lattice-order on groups of paraunitary matrices

TL;DR

This paper introduces a new right-invariant lattice-order on groups of paraunitary matrices, providing an explicit description of universal groups for certain orthomodular lattices derived from vector spaces with bilinear forms.

Contribution

It offers a novel lattice-order structure on paraunitary matrix groups and explicitly describes their universal groups for specific orthomodular lattices.

Findings

Explicit description of universal groups in terms of paraunitary matrices

Establishment of a right-invariant lattice-order on these groups

Connection between orthomodular lattices and paraunitary matrix groups

Abstract

In \cite{rump_goml}, Rump defined and characterized noncommutative universal groups $G(X)$ for generalized orthomodular lattices $X$. We give an explicit description of $G(X)$ in terms of \emph{paraunitary} matrix groups, whenever $X$ is the orthomodular lattice of subspaces of a finite-dimensional $k$-vector space $V$ that is equipped with an anisotropic, symmetric $k$-bilinear form.