Unitary Steinberg group is centrally closed
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A. Lavrenov
Translated by: the author - St. Petersburg Math. J. 24 (2013), 783-794
- DOI: https://doi.org/10.1090/S1061-0022-2013-01265-9
- 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.References
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Bibliographic 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
- Received by editor(s): May 22, 2012
- Published electronically: July 24, 2013
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 783-794
- MSC (2010): Primary 19C09
- DOI: https://doi.org/10.1090/S1061-0022-2013-01265-9
- MathSciNet review: 3087823