Another presentation for symplectic Steinberg groups

TL;DR

The paper proves that the symplectic Steinberg group over any commutative ring is a central extension of the elementary symplectic group, providing a new presentation that clarifies its structure and relation to symplectic K-theory.

Contribution

It introduces a new set of generators and relations for the symplectic Steinberg group, demonstrating its centrality and aligning explicit and implicit definitions of symplectic K_2.

Findings

Symplectic Steinberg group is a central extension for all rings and ranks ≥ 3.

New presentation makes the centrality of the extension obvious.

Explicit definition of symplectic K_2 matches the classical plus-construction approach.

Abstract

We solve a classical problem of centrality of symplectic $\mathrm K_2$, namely we show that for an arbitrary commutative ring $R$, $l\geq3$ the symplectic Steinberg group $\mathrm{StSp}(2l,\,R)$ as an extension of the elementary symplectic group $\mathrm{Ep}(2l,\,R)$ is a central extension. This allows to conclude that the explicit definition of symplectic $\mathrm{K_2Sp}(2l,\,R)$ as a kernel of this extension, i.e. as a group of non-elementary relations among symplectic transvections, coincides with the usual implicit definition via plus-construction. We proceed from van der Kallen's classical paper, where he shows an analogous result for linear K-theory. We find a new set of generators for the symplectic Steinberg group and a defining system of relations among them. In this new presentation it is obvious that the symplectic Steinberg group is a central extension.