Generalized Quadrangles Weakly Embedded of Degree 2 in Projective Space

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: (i) π is injective on points, (ii) if x ∈ P and L ∈ L with x I L, then x is incident with L in PG(V ), (iii) the set of points {xπ | x ∈ P} generates PG(V ), (iv) if x, y ∈ P such that y 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, (v) 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).