Equality of elementary symplectic group and symplectic group

TL;DR

This paper generalizes Kopeiko's result by proving that over Euclidean rings, the symplectic group defined with respect to any invertible skew-symmetric matrix of Pfaffian one coincides with the elementary symplectic group.

Contribution

It extends Kopeiko's theorem to symplectic groups defined by arbitrary invertible skew-symmetric matrices with Pfaffian one.

Findings

Symplectic group equals elementary symplectic group for any invertible skew-symmetric matrix of Pfaffian one.

Generalization of Kopeiko's result to broader class of symplectic groups.

Provides a unified understanding of symplectic groups over Euclidean rings.

Abstract

V.I. Kopeiko proved that over a euclidean ring, the symplectic group defined with respect to the standard skew-symmetric matrix is same as the elementary symplectic group. Here we generalise the result of Kopeiko for a symplectic group defined with respect to any invertible skew-symmetric matrix of Pfaffian one.