Visualization for Petrov's Odd Unitary Group

TL;DR

This paper introduces a new set of matrices related to Petrov's odd unitary group, proves their elementary nature, and explores their role within the group's structure, extending classical group concepts.

Contribution

It defines matrices analogous to Vaserstein-type matrices, proves their elementary status, and shows their relation to Petrov's odd unitary group in a novel way.

Findings

Matrices are elementary linear matrices.

Under certain conditions, matrices belong to Petrov's odd unitary group.

Generated group is a conjugate of Petrov's odd elementary hyperbolic unitary group.

Abstract

In this article we define a set of matrices analogous to Vaserstein-type matrices which was introduced in the paper `Serre's problem on projective modules over polynomial rings and algebraic K-theory' by Suslin-Vaserstein in 1976. We prove that these are elementary linear matrices. Also, under some conditions, these matrices belong to Petrov's odd unitary group which is a generalization of all classical groups. We also prove that the group generated by these matrices is a conjugate of the Petrov's odd elementary hyperbolic unitary group when the ring is commutative.