Generalised homotopy and commutativity principle

TL;DR

This paper investigates the action of certain linear and symplectic matrices homotopic to the identity on right invertible matrices, and establishes properties of the commutator subgroup of orthogonal groups over polynomial rings.

Contribution

It introduces a generalized homotopy and commutativity principle for linear and symplectic matrices, and proves the stable elementary nature of the commutator subgroup in orthogonal groups.

Findings

Action of special matrices on right invertible matrices characterized

Commutator subgroup of orthogonal groups shown to be two stably elementary

Results applicable over local rings with 1/2

Abstract

In this paper, we study the action of special $n\times n $ linear (resp. symplectic) matrices which are homotopic to identity on the right invertible $n\times m$ matrices. We also prove that the commutator subgroup of $\rm{O}_{2n}(R[X])$ is two stably elementary orthogonal for a local ring $R$ with $\frac{1}{2}\in R$ and $n\geq 3.$