Unitary Steinberg group is centrally closed

Abstract:

Let $(R,\Lambda )$ be an arbitrary form ring, let $U(2n,R,\Lambda )$ denote the hyperbolic unitary group, let $EU(2n,R,\Lambda )$ be its elementary subgroup and $\mathrm {StU} (2n,R,\Lambda )$ the unitary Steinberg group. It is proved that, if $n\ge 5$ (a natural assumption for similar results), then every central extension of $\mathrm {StU} (2n, R,\Lambda )$ splits. This results makes it possible to describe the Schur multiplier of the elementary unitary group as the kernel of the natural epimorphism of $\mathrm {StU}(2n, R, \Lambda )$ onto $EU (2n, R,\Lambda )$ if it is known that this kernel is included in the center of the unitary Steinberg group. Steinberg’s description of relations is employed, which leads to simplest proofs of these results.