Another presentation of orthogonal Steinberg groups

TL;DR

This paper demonstrates a new presentation of orthogonal Steinberg groups using the pro-group approach and constructs analogs of ESD-transvections under certain conditions.

Contribution

It introduces a novel presentation for orthogonal Steinberg groups and constructs ESD-transvections analogs within the pro-group framework.

Findings

Established van der Kallen's presentation for orthogonal Steinberg groups.

Constructed ESD-transvections analogs in orthogonal Steinberg pro-groups.

Extended the understanding of the structure of orthogonal Steinberg groups.

Abstract

We use the pro-group approach to show that $\mathrm{StO}(M, q)$ admits van der Kallen's "another presentation", where $M$ is a module over a commutative ring with sufficiently isotropic quadratic form $q$. Moreover, we construct an analog of ESD-transvections in orthogonal Steinberg pro-groups under some assumptions on their parameters.