Moufang sets arising from polarities of Moufang planes over octonion division algebras

摘要

For every octonion division algebra O, there exists a projective plane whichis parametrized by O; these planes are related to rank two forms of linearalgebraic groups of absolute type E6. We study all possible polarities of suchoctonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving riseto (1) Moufang sets of type F4, (2) Moufang sets of type 2E6, (3) hermitianMoufang sets of type C4, and (4) projective Moufang sets over a 5-dimensionalsubspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two,whereas case (4) only occurs over fields of characteristic equal to two. TheMoufang sets of type 2E6 that we obtain in case (2) are exactly thosecorresponding to linear algebraic groups of type 2E6,1^29; the explicitdescription of those Moufang sets was not yet known.