UNITARY STEINBERG GROUP IS CENTRALLY CLOSED

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 StU(2n, R, Lambda) the unitary Steinberg group. It is proved that, if n >= 5 (a natural assumption for similar results), then every central extension of 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 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.