On the embeddability of polar spaces

Abstract

We show that every nondegenerate polar space of rank at least 4 with at least three points on each line can be embedded in a projective space. Together with some results from [9] and 1-12], this provides a particularly elementary proof that any such polar space is of classical type. Our methods involve the use of geometric hyperplanes and work equally well for spaces of finite or infinite rank.

On Polar Spaces of Infinite Rank

“…As proved by Tits [17, chapters 8 and 9] (see also Buekenhout and Cohen [1, chapters 7-11] and Cuypers et al [4]) all polar spaces of rank at least 3 are embeddable but for two families of polar spaces of rank 3. The non-embeddable ones are the line-grassmannians of 3-dimensional projective spaces defined over non-commutative division rings and certain polar spaces with Moufang but non-desarguesian planes, described in [17,Chapter 9].…”

The geometry of hyperbolic lines in polar spaces

“…If the rank of the polar space is infinite, then it is embeddable in a projective space such that hyperbolic lines are contained in projective lines, see [13,25,32]. Assume we have such an embedding of (P, L).…”

Abstract Root Subgroups and Simple Groups of Lie-Type

“…If rk(P(E)) ~ 3 then it follows from the classification of polar spaces by Buekenhout-Shult [BS74] , Tits-Veldkamp [Tit74] and Cuypers, Johnson and Pasini [CJP92] in the infinite rank case that either P(E) is classical, i.e. isomorphic to the polar space obtained from isotropic points and lines of some vector space with some form or is isomorphic to the uniquely determined polar space of rank 3, the planes of which are Moufang planes over some Cayley division algebra K. For details see [Coh95].…”