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Quadratic maps and Bockstein closed group extensions

Authors:	Jonathan Pakianathan Ergü n Yalç in

Institution:	Department of Mathematics, University of Rochester, Rochester, New York 14627 ; Department of Mathematics, Bilkent University, Ankara, 06800, Turkey

Abstract:	Let be a central extension of the form $0 \to V \to G \to W \to 0$ where and are elementary abelian -groups. Associated to there is a quadratic map $Q: W \to V$ , 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 $H^2(G, \mathbb{Z} /2)$ 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 $Q_{\mathfrak{gl}_n}: \mathfrak{gl}_n (\mathbb{F}_2)\to \mathfrak{gl}_n (\mathbb{F}_2)$ given by $Q(\mathbb{A})= \mathbb{A} +\mathbb{A} ^2$ 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 $0 \to M \to \widetilde{G} \to W \to 0$ for some $\mathbb{Z} /4W]$ -lattice . In this situation, one may write $\beta (q)=Lq$ for a ``binding matrix' with entries in $H^1(W, \mathbb{Z}/2)$ . 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.

Keywords:	Group extensions quadratic maps group cohomology restricted Lie algebras

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