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Let 
be the exterior algebra on 
, and let 
be
the subspace of 
-vectors of . Then 
and the
decomposable 2-vectors are precisely those vectors in
whose plückerian
coordinates
satisfy the quadratic equation 
, where 
is the matrix 
. This defines a quadric 
in 6-space, the
famous Klein's quadric. Since decomposable 2-vectors correspond to
2-dimensional
subspaces of 
, the points of 
represent in the projective space
the lines of the projective space 
.
Under the above correspondence every ``ruled" geometric object in is
represented
by a set of points on in . Thus a linear complex 
is
represented
by the intersection of a hyperplane 
of with (cf.
2.1). will be tangent to 
if is special, otherwise
will be a non-degenerate quadric of 
. As for a linear
congruence of 
, one has to intersect the hyperplanes
representing the corresponding complexes and then intersect this 3-dimensional
subspace of 
with 
. Thus a linear congruence 
is represented
by an ``ordinary" quadric lying on 
. According to being
degenerate, parabolic, hyperbolic or elliptic, the quadric will be degenerate
into two planes, a cone, or a non-degenerate quadric of index 2 and 1
respectively. In the same way, a regulus of a ruled quadric of can be
seen as the intersection with 
of a plane of , i.e. as a conic
lying on 
.
Let 
be the form defining the Klein's quadric 
, let 
and 
be the
exterior square of 
. Then, for every
,
.
Thus the map 
induces a homomorphism from 
to
.
Let 
, 
be the projective transformations corresponding to
and 
respectively. It is easy to check that 
iff 
is a square. It follows that 
is isomorphic
to the subgroup of 
consisting of the elements having determinant a
square. From this we get an isomorphism between 
and 
.
Now 
preserves the non-special linear complex 
consisting of the
isotropic lines of , which is represented in by the
intersection of a hyperplane with 
. This is
a non-degenerate quadric of 
. Hence we get a monomorphism of
into 
, i.e. an isomorphism between and 
.
(For details see [VdW] and [Dieu].)
Next: The maximal subgroups
Up: Some geometryand
Previous: Linear complexes and