Quadratic maps and Bockstein closed group extensions
Authors: Jonathan Pakianathan and Ergün Yalçin
Journal: Trans. Amer. Math. Soc. 359 (2007), 6079-6110
MSC (2000): Primary 20J05; Secondary 17B50, 15A63
Published electronically: May 7, 2007
MathSciNet review: 2336317
Abstract: Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.
We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.
On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix'' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.
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Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Affiliation: Department of Mathematics, Bilkent University, Ankara, 06800, Turkey
Keywords: Group extensions, quadratic maps, group cohomology, restricted Lie algebras.
Received by editor(s): December 2, 2005
Published electronically: May 7, 2007
Additional Notes: The second author was partially supported by a grant from the Turkish Academy of Sciences (TÜBA-GEBİP/2005-16).
Article copyright: © Copyright 2007 American Mathematical Society