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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Unitary Steinberg group is centrally closed


Author: A. Lavrenov
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 783-794
MSC (2010): Primary 19C09
Published electronically: July 24, 2013
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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.


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Additional Information

A. Lavrenov
Affiliation: Department of mathematics and mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: avlavrenov@gmail.com

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01265-9
PII: S 1061-0022(2013)01265-9
Keywords: Unitary Steinberg group, Schur multiplier, unitary group, form parameter, nonstable $K$-theory
Received by editor(s): May 22, 2012
Published electronically: July 24, 2013
Article copyright: © Copyright 2013 American Mathematical Society