With a G-null basis for Hilb(n) we can construct a G-null basis for the algebra Alg(n) of linear operators on Hilb(n). If |v_k> is such a basis then

        V_kj  =  |v_k><v_j|

   spans Alg(n) as a vector space and

        Tr( V_kj G(n) )  =  0

   for all k,j = 0, 1, 2, ..., n-1.
   Then V_kj is basis for Alg(n) and is G-null with respect to the
   pseudometric induced on Alg(n) from by Hilb(n).

   The Geometry of the G(2)-null cone: In C^2

        |z_1|^2 - |z_2|^2  =  0
   or
        r_1  =  r_2

   with 2 unconstrained phases.  As a surface in R^4

        (x_1^2 + y_1^2) - (x_2^2 + y_2^2)  =  0

        (x_1^2 + y_1^2)  =  (x_2^2 + y_2^2)

Artificially make one of the radii very large. On this large circle attach at every point a copy of the other circle whose plane is orthogonal to the tangent vector of the large circle at the point of attachment. We have generated a torus whose centroid is the large circle. If the large circle radius is contracted to match that of the smaller circle, the doughnut hole of the torus has just collapsed to a point. Thus generate the resulting 3-surface by drawing a circle (the first T^1) and its tangent at a single point. Revolve the circle about the tangent line. (The center of the first T^1 traces out the second T^1.) The swept out surface is the 2-dimensional surface (T^1 X T^1), while its size r is the third parameter. Either of the T^1 together with r determines a cone.

Consider the 3-dimensional solid of which (T^1 X T^1) is the boundary. Before contracting the large radius, say r1, to that of the small radius, so r_1 > r_2, (T^1(r_1) X T^1(r_2)) is a genuine doubly connected T^2. If r_1 < r_2, it looks like the T^2 has lost its double connectedness. Not so, the two complex planes (which contain the the individual T^1's) enter perfectly symmetrically in C^2. What has happened is that a particular projection into an R^3 has lost double connectedness. The single point closure of the doughnut hole is not sufficient to free paths that cross the "interior equator of torus". One has still two classes of paths not continuously deformable to each other: class 1 = paths that are deformable to the closure point, class 2 = paths that are not deformable to the closure point. If a path (that is not a point) goes through the closure point, it is a class 2 path.

        --------------------------------------------------------------
        The 2-dimensional cross section of the 3-dimensional G(2)-null
        cone is doubly connected torus with a central hole contracted
        to a point.
        --------------------------------------------------------------

Question: A normal 2-dim toroidal surface, unlike the 2-dim sphere, can be covered by a single coordinate patch with no singularities. The toroidal cross section of the G(2)-null cone, appears to have a singularity much like the northpole singularity for the sphere. If the closure point is split into two and pulled apart, the result is toplogically equivalent to a sphere with a latitude and longitude coordinate system with two singularities.

Can one say from this that the G(2)-null cone, is essentially the same topologically (modulo the point surgery) as the Minkowski light cone? Strictly speaking, of course, without the point surgery, there are still two homotopy classes of paths, and so the the topology is that of a toroid.

   Project the z_2-plane to a line: z_2 -> r_2
        (x_1^2 + y_1^2)  =  r_2^2
   which is the equation of a cone in an R^3 as r_2 varies.
        sqrt(x_1^2 + y_1^2)  =  r_2


   Similarly project the z_1-plane to a line: z_1 -> r_1
        (x_2^2 + y_2^2)  =  r_1^2
   which is the equation of a cone in an R^3 as r_1 varies.
        sqrt(x_2^2 + y_2^2)  =  r_1

   If (z_1, z_2) is a cone point
   (wz_1, wz_2), w not= 0 is also a cone point.


   The Geometry of the G(3)-null cone: In C^3

        |z_1|^2 + |z_2|^2 - |z_3|^2  =  0
   or
        r_1^2 + r_2^2  =  r_3^2

   with 3 unconstrained phases.  As a surface in R^6

        (x_1^2 + y_1^2) + (x_2^2 + y_2^2) - (x_3^2 + y_3^2)  =  0

        (x_1^2 + y_1^2) + (x_2^2 + y_2^2)  =  (x_3^2 + y_3^2)

If the LHS is contant, this describes a 3-dimensional spherical surface in an R^4. If the RHS side is constant a T^1 is described. Again as for n=2, the null cone

   is 2n-1 = 5 dimensional surface in R^2n

   Grossly characterized by

          [ 2n-3 dimensional cross section ] X [ size parameter ]

   The cross section has the topology: S^(2n-3) X T^1 (S^4 X T^1)
   while the size parameter has obvious topology R^1.

   Generalizing, the topology of a (Lorentzian) G(n)-null cone in Hilb(n)
   is

        N_1(C, n)  <->  R X [ S^(2n-3) X T^1 ]

   where [ S^(2n-3) X T^1 ] is the cross section.


   Comparing a real Lorentzian null-cone N_1(R, n):

        N_1(R, n):
		Dimension: n-1
		Topology: (R^1 > 0) X S^(n-2)  +  (R^1 < 0) X S^(n-2)
		          forward cone            backward cone
			2 Disconnected pieces
                Cross section Topology: S^(n-2)
			Connected and simply connected.
     N_1(CK, n):
		Dimension: 2n-1
		Topology: (R^1 > 0) X [ S^(2n-3) X T^1 ]
		          Forward cone & backward cone are continuously
			  connected (through T^1).
                Cross section Topology: S^(2n-3) X T^1
			Connected and doubly connected.

   For null cones in spaces with signature 1 < s < n-s,
   other than Lorentian (s=1)
   NB:
        S^1 = T^1

        N_s(R, n):
		Dimension: n-1
		Topology: (R^1 > 0) X S^(s-2) X S^(2n-s-2)
		          Forward cone & backward cone are continuously
			  connected (through S^(s-1).
                Cross section Topology: S^(s-1) X S^(n-s-2)
			Connected and doubly connected.
			2 classes of paths.
     N_s(C, n):
		Dimension: 2n-1
		Topology: (R^1 > 0) X [ S^(2n-2s-2) X S^(2s-1) ]
		          Forward cone & backward cone are continuously
			  connected (through T^(2s-1).
                Cross section Topology: S^(2n-2s-2) X S^(2s-1)
			Connected and doubly connected.
			2 classes of paths.


   Equalities are homeomorphisms

			     N_s(R, n) = N_n-s(R, n)
			     N_s(C, n) = N_n-s(C, n)

     N_1(C, 2)  =  N_2(R, 4)
     N_1(C, 3)  =  N_2(R, 6)
     N_1(C, 4)  =  N_2(R, 6)
              ... 
     N_1(C, n)  =  N_2(R, 2n)


     N_2(C, 3)  =  N_1(C, 3)  =  N_2(R, 5)  =  N_3(R, 5)
     N_2(C, 4)                =  N_4(R, 8)
     N_2(C, 5)  =  N_3(C, 5)  =  N_4(R, 10) =  N_6(R, 10)

              ...

			     N_s(C, n)  =  N_2s(R, 2n)



            The Complex Projective Space P_n(C)

The complex Projective space can be defined as a homogeneous factor space:

        P_n(C)  =  SU(n+1)/U(n)

   Explicitly construct P_n(C) as follows:

   A point of C^(n+1) is an (n+1)-tuple of complex numbers

             z = (z_0, z_1, ..., z_n).

   The elements of P_n(C) are equivalence classes of (n+1)-tuples where
   z is said to be equivalent to w iff for some nonzero complex number
   lambda,

             z  =  lambda w

             w_k  =  z_k/z_n

             f(lambda z)  =  lambda^r f(z)

                S^2 X S^2 X ... X S^2
                ---------------------
                       PI^(n-1)

          |psi_s>  =  SIGMA |q(n, k)>
                         k

        <psi_s|G(n)|psi_s>

        =    SIGMA  ( delta_(kj) - (-1)^(k-j) )
              k,j

   where k,j = 0, 1, ..., n-1

        =   - SIGMA  (-1)^(k-j)
            k not= j

        =   2 nu

   for n = 2 nu and n = 2 nu + 1.

and is therefore in Hilb_+(n)

The complex Hyperbolic space can be defined as a homogeneous factor space [Appendix D]:

               H_n(C) = SU(n, 1)/U(n)

          <psi| G(n) |psi>  =  -r^2

          {{exp( i theta )|psi>}: <psi| G(n) |psi>  =  -r^2}

   For the vector (Cf.  Lemma 8.6.)

                    (n-1)
        |psi_->  =  SIGMA  (-1)^k r_k |q(n, k)>
                     k=0

   with r_k > 0,

        <psi_-|G(n)|psi_->

        =    SIGMA  ( delta_kj - (-1)^(k-j) (-1)^(k+j) r_k r_j )
              k,j

        =   - SIGMA  r_k r_j  < 0
            k not= j

   For the vector

                    (n-1)
        |psi_->  =  SIGMA  (i)^k |q(n, k)>
                     k=0

        <psi_-|G(n)|psi_->

        =  - SIGMA  exp( i (pi/2) (k-j) )
            k not= j

        =  - SIGMA  cos( i( pi/2)(k-j) )
            ^(knot=j

           |  -(n-1),         n  =  2nu + 1
        =  |  -(n-2),         n  =  2nu, and nu is odd
           |    -n,           n  =  2nu, and nu is even

        <  0


   Therefore both vectors are in Hilb_-(n).