Another presentation for symplectic Steinberg groups

We solve a classical problem of centrality of symplectic K2K2, namely we show that for an arbitrary commutative ring R  , l≥3l≥3, the symplectic Steinberg group StSp(2l,R)StSp(2l,R) as an extension of the elementary symplectic group Ep(2l,R)Ep(2l,R) is a central extension. This allows to conclude that the explicit definition of symplectic K2Sp(2l,R)K2Sp(2l,R) as a kernel of the above 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.