Quadrangles embedded in metasymplectic spaces

This chapter proves various results about Moufang quadrangles. It first considers the notions of a proper involutory set, a proper indifferent set, and a proper anisotropic pseudo-quadratic space. It then shows that the root group sequence Ω‎ is isomorphic to a root group sequence of exactly one of six types relating to some proper involutory set, some non-trivial anisotropic quadratic space, some proper indifferent set, some proper anisotropic pseudo-quadratic space, and some quadratic space. It also describes the degree of a finite purely inseparable field extension as a power of the characteristic, an isomorphism from a root group sequence of Δ‎ to the Moufang quadrangle, and abelian and non-abelian groups.

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is iden- tical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F_4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons with n ≥ 3.

Akihiro Munemasa, Christos Athanasiadis, Hugh Thomas, and Hendrik van Maldeghem share the chief editor role at a journal that's like many others across mathematics and the sciences.