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On the Odd Unitary Analogue of Gram-Schmidt Process

Ambily A. A., Aparna Pradeep V. K

TL;DR

The paper addresses whether a Gram-Schmidt-type construction in Petrov's odd unitary framework can generate all elementary matrices. It develops Vaserstein-type matrices via $\theta(v)$ and $\eta(v)$, built from a fixed form $\Psi=\widetilde{\psi}_{m} \perp \varphi$, and proves these matrices generate the full elementary group ${\rm E}_{n+2m-1}(R)$ for arbitrary $\varphi$. The main contribution is a general, direct proof that the subgroup generated by $\theta(v)$ and $\eta(v)$ coincides with ${\rm E}_{n+2m-1}(R)$, extending previous Pfaffian-1 restricted results. This work broadens the Gram-Schmidt-type generation to the broader setting of Petrov's odd unitary groups, facilitating uniform proofs of classical results and potential applications to stability and $K_1$-type questions in this context.

Abstract

In 1976, L.N. Vaserstein used a construction analogous to the Gram-Schmidt orthogonalisation, for obtaining a set of symplectic matrices from a set of elementary matrices. We have a similar construction for Petrov's odd unitary group. Here, we prove that the elementary matrices in the odd unitary analogue of the Gram-Schmidt process form a set of generators for the elementary linear group.

On the Odd Unitary Analogue of Gram-Schmidt Process

TL;DR

The paper addresses whether a Gram-Schmidt-type construction in Petrov's odd unitary framework can generate all elementary matrices. It develops Vaserstein-type matrices via and , built from a fixed form , and proves these matrices generate the full elementary group for arbitrary . The main contribution is a general, direct proof that the subgroup generated by and coincides with , extending previous Pfaffian-1 restricted results. This work broadens the Gram-Schmidt-type generation to the broader setting of Petrov's odd unitary groups, facilitating uniform proofs of classical results and potential applications to stability and -type questions in this context.

Abstract

In 1976, L.N. Vaserstein used a construction analogous to the Gram-Schmidt orthogonalisation, for obtaining a set of symplectic matrices from a set of elementary matrices. We have a similar construction for Petrov's odd unitary group. Here, we prove that the elementary matrices in the odd unitary analogue of the Gram-Schmidt process form a set of generators for the elementary linear group.

Paper Structure

This paper contains 4 sections, 3 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Odd Unitary Groups
  4. Vaserstein-type matrices for odd elementary hyperbolic unitary group

Key Result

Theorem 1.1

ChattopadhyayRao2024 Let $\varphi$ be an alternating matrix of Pfaffian 1 of size $2n$, with $n \geq2$. Then the groups ${\rm E}_{2n-1}(R)$ and ${\rm E}_{\varphi}(R)$ are equal.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: odd unitary group
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Remark 3.2