Abstract

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 in P and L in L with x I L, then x^π is incident with L^π in PG(V ),
• the set of points {x^π | x in P} generates PG(V ),
• if x,y in P such that y^π is contained in the subspace of PG(V ) generated by the set {z^π | 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 P^π 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).

Authors