On Polar Spaces of Infinite Rank

Abstract

Many properties of polar spaces of finite rank fail to hold in polar spaces of infinite rank. For instance, in a polar space of infinite rank it can happen that maximal singular subspaces have different dimensions; every polar space of infinite rank contains singular subspaces that cannot be obtained as intersections of any family of maximal singular subspaces, whereas in a polar space of finite rank every singular subspace is the intersection of a finite number of maximal singular subspaces. In this paper we shall examine peculiar properties of polar spaces of infinite rank, to test how far them can drug us from the familiar world of polar spaces of finite rank.

Sets of generators and chains of subspaces

“…Therefore dim(e) ≥ rk gen (Γ). Hence dim(e) = rk gen (Γ) by (7). ✷ Proposition 4.5 Every faithful embedding is dominant.…”

“…Remark 6 In [7,Introduction] it is wrongly claimed that (5) holds in general. However, no mention of the number prk C (Γ) is made in [7] after that claim. So, luckily, that error has no consequences in [7].…”

“…However, no mention of the number prk C (Γ) is made in [7] after that claim. So, luckily, that error has no consequences in [7].…”

“…Remark 5 When Γ admits singular subspaces of infinite rank, all maximal singular subspaces of Γ have infinite rank but not necessarily the same rank (see e.g. [7]). Clearly, the polar rank prk(Γ) defined as in ( 4) is the least upper bound (but possibly not the maximum) of the ranks of the maximal singular subspaces of Γ.…”

An outline of polar spaces: basics and advances

“…Note that the strong separation properties are proved true also for nondegenerate polar spaces of infinite rank provided there exists a maximal singular subspace of countable dimension (see Pasini [41]).…”