In this paper, we classify
all generalized quadrangles weakly embedded of degree 2 in projective space. More
exactly, given a (possibly infinite) generalized quadrangle Γ = (𝒫,ℒ,I) and a map π
from 𝒫 (respectively ℒ) to the set of points (respectively lines) of a projective space
PG(V ), V a vector space over some skew field (not necessarily finite-dimensional),
such that:
π is injective on points,
if x ∈𝒫 and L ∈ℒ with x I L, then xπ is incident with Lπ in PG(V ),
the set of points {xπ∣x ∈𝒫} generates PG(V ),
if x,y ∈𝒫 such that yπ is contained in the subspace of PG(V ) generated
by the set {zπ∣z is collinear with x in Γ}, then y is collinear with x in Γ,
there exists a line of PG(V ) not in the image of π and which meets 𝒫π in
precisely 2 points,
then we show that Γ is a Moufang quadrangle and we can explicitly describe the weak
embedding of Γ in PG(V ). This completes the classification of all weak
embeddings of arbitrary generalized quadrangles (using the classification of
Moufang quadrangles).
