Projective Points Over Matrices and Their Separability Properties

In this article we consider topological quotients of real and complex matrices by various subgroups and their connections to spacetime structures. These spaces are naturally interpreted as projective points. In particular, we look at quotients of nonzero matrices M2∗(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^*_2({\mathbb {F}})$$\end{document} by GL2(F),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$GL_2({\mathbb {F}}),$$\end{document}SL2(F),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_2({\mathbb {F}}),$$\end{document}O2(F),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_2({\mathbb {F}}),$$\end{document} and SO2(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO_2({\mathbb {F}})$$\end{document} and prove various results about their topological separability properties. We discuss the interesting result that, as the group we quotient by gets smaller, the separability properties of the quotient improve.


Introduction
There exists a substantial literature on the relationship between various algebras (real division algebras, Clifford algebras, etc.) and relativistic physics, [7][8][9]. Matrix algebras play a fundamental role, and in this article we extend the connection by considering topological quotients of matrix algebras. The resulting spaces, however, are non-Hausdorff. Such spaces are not as well studied as their Hausdorff counterparts, and for good reason: Hausdorff separability is prerequisite for many other properties, such as being locally Euclidean, that are essential in much of mathematics and its applications. But this is not the case in all of mathematics and its applications, and indeed we observe a growing interest in spaces which encode information by way of the lack of separability properties. Applications of these more "exotic" spaces continue to appear in areas as diverse as category theory, algebra, operator algebra, algebraic geometry, logic, computer science, and mathematical physics (see, e.g., [1,4,10,11,13,19] for a sampling).
In a prior article [2], the first author followed up on results by Souček [16,17] regarding the connection between a projective space built over a matrix ring and Penrose's Twistor Theory [12,14,15]. It was subsequently recognized that a large class of such projective spaces over matrices could be constructed and analyzed in the same spirit, yielding generalizations of the fundamental twistor correspondence [3]. These matrix projective spaces organize and unify the Grassmannians of the underlying vector space in such a way that the incidence properties of subspaces are encoded into the (lack of) separation properties of the parent matrix projective space. In that paper [3], the nonzero matrices were quotiented by the subset of invertible matrices, as that was the context. (It is worth noting that other interesting studies of projective spaces over rings have been done, such as in [6,18], but our construction differs in an essential way that leads to nonhomogeneous spaces.) Inspired by these results, we found it interesting to consider the effect of quotienting by other natural subgroups, particularly those that have known significance and applications to physics and geometry. This article presents the results of a first foray into this territory for the case of real and complex 2 × 2 nonzero matrices quotiented by the most common Lie subgroups (the general and special linear, and the regular and special unitary/orthogonal groups). We are able to obtain a complete description of these spaces and their topologies, and we find an interesting (and reasonable) relationship between the separation properties and the size of the group we are quotienting by.
It will clarify the exposition to take each topological quotient in turn. In preparation for this, let F denote the real or complex numbers, M 2 (F) the space of 2 × 2 matrices over F, and M * 2 (F) the subspace of nonzero matrices. We will also be interested in the general linear group GL 2 (F), the special linear subgroup SL 2 (F), the orthogonal subgroup O 2 (F), and the special orthogonal subgroup SO 2 (F). In the case F = C, of course, the orthogonal group is called the unitary group, and it and its special counterpart are denoted U (2) and SU (2), respectively.
Let D(F) denote any one of GL 2 (F), SL 2 (F), O 2 (F), or SO 2 (F). The construction of the quotient spaces P 0 D (F) := M * 2 (F)/D(F) largely follows the standard construction of a projective space of dimension k over a field: 1. Form the free F-module of rank k + 1; 2. Throw out the zero element (and anything else as dictated by the context); 3. Define equivalence classes modulo scalar multiplication by some set of invertible elements. The projective space is then the set of equivalence classes endowed with the quotient topology. When dealing with scalars that come from a noncommutative ring, we must choose between right and left scalar multiplication. The difference is one of duality and will not concern us here (see [2]). In the present article, we will scalar multiply on the right, throw out only the zero element, consider a few different sets of scalars for our quotienting set, and set k = 0 (the projective point case).
For convenience, we recall the three weakest of the separation axioms.
Definition. We say that a point P is T 0 -separable from a point Q if there exists a neighborhood of P excluding Q. We say that a pair of points is T 0 -separable if at least one of the points is T 0 -separable from the other. A topological space is said to be T 0 (or Kolmogorov ) if every pair of distinct points is T 0 -separable.
Definition. We say that a pair of points is T 1 -separable if each point has a neighborhood excluding the other. A topological space is said to be T 1 (or Fréchet) if every pair of distinct points is T 1 -separable. We note that a space is T 1 if and only if every singleton set is closed.
Definition. We say that a pair of points is T 2 -separable if each point has a neighborhood excluding the other such that the two neighborhoods are disjoint. A topological space is said to be T 2 (or Hausdorff ) if every pair of distinct points is T 2 -separable.

Construction and Analysis of the Case D = GL
The first space we wish to draw attention to is P 0 GL (F). It is the instance with the largest quotienting set. The case that F = C was studied for other purposes in [2]. Here we will summarize what we need for the present discussion, appropriately generalized for either C or R.
We are looking at equivalence classes of M * 2 (F) modulo right multiplication by elements of GL 2 (F) : A singular nonzero matrix, A, will have rank one, and thus the columns a 1 , a 2 will be multiples: λ a μ a , where not both λ and μ are zero. By using elementary matrices in GL 2 (F) we can zero out the second column, and then the remaining freedom consists of identifying nonzero multiples: Thus, the equivalence classes deriving from the nonzero singular matrices are bijective with the projective line, FP 1 . In particular, CP 1 ∼ = S 2 , and On the other hand, all invertible elements are identified by this relation to form a single equivalence class * . What is interesting is the way that this class interacts with the rest of the quotient.
The topology induced by the quotient mapping q : M * 2 (F) → P 0 GL (F) on the subspace of equivalence classes with singular representative is the standard topology on FP 1 , while any neighborhood of a singular matrix will contain nonsingular matrices, so the projective space P 0 GL (F) is homeomorphic to the disjoint union of a 2-sphere and a dense point when F = C, and The separation properties, closed points, and closures of non-closed points of these spaces are detailed in the following proposition.

A singleton set containing a point [A]
∈ FP 1 is closed, while the closure of the dense point is P 0 GL (F). Proof. Since * is dense, the closure of the singleton { * } is the whole space, and so P 0 GL (F) is not T 1 . We can see that P 0 GL (F) is T 0 by looking at basic open sets in the parent space (cf. Fig. 2) and noting that the subspace topology on the singular classes is just the usual topology of FP 1 : Every open set of M * 2 (F) will intersect GL 2 (F), so there is no way to separate any other point of P 0 GL (F) from * . However, * can be separated from any other point by the open set { * }. Any two distinct singular classes can be T 1 -separated by using separating neighborhoods in the FP 1 topology and adding the dense point, but any two such neighborhoods will overlap in the dense point.

Construction and Analysis of the Case D = SL
We now consider the construction of P 0 SL (F) using the same procedure above, but now we quotient by the smaller group SL 2 (F). It will be convenient once again to separate out the nonsingular and singular cases.
It turns out that in the construction of P 0 GL (F), in the singular case, only unit determinant matrices are needed to get unique equivalence class representatives in bijection with The curve represents the nonzero singular matrices Sing(2, F). Any neighborhood of a singular matrix must contain nonsingular matrices, so that no equivalence class can be separated from the nonsingular class * . The quotient for any non-zero scalar α via the squeeze transformation α 0 0 α −1 . Thus, equivalence classes are represented by non-zero vectors in F 2 up to scalar multiplication, resulting again in FP 1 , in P 0 SL (F). For the nonsingular case we have the following lemma, which follows directly from standard results in linear algebra: Lemma. The equivalence classes of P 0 SL (F) having nonsingular representatives are in bijection with the nonzero scalars, F * , each class represented by the determinant of any representative matrix.
So, the difference between this case and the case D = GL is that the dense point in the GL case "blows up" into a punctured plane (F = C) or punctured line (F = R) when we quotient only by SL matrices. Or, in reverse, when we step up from a quotient of SL matrices to a quotient of GL matrices, all the points of F * get identified. The topological interest lies, again, in the interaction between singular classes and the rest of the quotient space. First, we note that the set of nonsingular matrices is open in M * 2 (F), so that in the quotient topology of P 0 SL (F), F * is open and FP 1 is closed. The separability properties are detailed in the following proposition.
Every singleton set is closed.  The last statement follows from the equivalent reformulation of a T 1 space.
Therefore, we may write The topology is such that, individually, the punctured-R i part carries the subspace topology inherited from the standard R i topology, and the S i part a standard S i topology (for i = 1 or 2). However, as parts of the quotient, the S i is "infinitely close" to the puncture in R i . Neighborhoods of distinct elements in S i necessarily overlap in the region of the puncture (see Fig. 3).

Construction and Analysis of the Case
and so P A = P B . Conversely, let A, B, and their polar decompositions be given. If P A = P B , then we may define Λ = U −1 B U A . Then, Thus, over C, the quotient is equivalent to the set of positive semidefinite Hermitian matrices, and over R, to the positive semi-definite symmetric matrices. It is straightforward to check that these correspondences are, in fact, homeomorphisms.
We will write where H ≥0 (resp., S ≥0 ) denotes the space of positive semi-definite Hermitian (resp., symmetric) matrices, H >0 (resp., S >0 ) denotes the space of positivedefinite Hermitian (resp., symmetric) matrices, which corresponds to the subspace of equivalences classes of nonsingular matrices, and (H ≥0 −H >0 ) (resp., (S ≥0 − S >0 )) corresponds to the subspace of equivalence classes of singular matrices. To see this, note that A = P A U A is singular if and only if P A is singular, since U A ∈ O 2 (F) is invertible.
This is a homeomorphism between real Minkowski vectors and Hermitian matrices [12]. Notice that where η is the Lorentzian inner product. That is, the Ψ map sends a real Minkowski vector to a Hermitian matrix with determinant equal to the squared length of the original vector. Positive semi-definite 2 × 2 matrices have non-negative determinant and positive trace, which implies that the corresponding spacetime vectors must have non-negative length and strictly positive time component, V 0 > 0. So, for F = C, positive semi-definite Hermitian matrices correspond to real future pointing timelike and lightlike vectors. We denote this space as As a subspace of RM 4 , the space V consists of the future pointing light cone minus the origin, corresponding to singular classes in P 0 O (F), along with its interior, corresponding to nonsingular classes. (See Fig. 4.) Moreover, for F = R, positive semi-definite symmetric matrices correspond to the subspace of V with V 2 = 0, the same real future-pointing timelike and lightlike vectors in a real Minkowski space one dimension smaller, RM 3 . We will denote this as Then we have an alternative formulation of the quotient: Topologically speaking, passing from the O 2 (F) quotient to the GL 2 (F) quotient amounts to collapsing the punctured light-cone to the compact space FP 1 , and collapsing the entire interior of that cone to a single point. Or again,   Proof. From the discussion above, P 0 O (F) is homeomorphic to a subspace of the metric space RM 4 .

Construction and Analysis of the Case D = SO
Finally, we consider the quotient by SO 2 (F), commonly denoted SU (2) over C and SO(2) over R. It will turn out to be quite closely related to the O 2 (F) quotient. Much like the SL quotient refines the GL quotient by the determinant, so too will the SO quotient refine the O quotient by the determinant. To this end, note that any 2 × 2 unitary matrix can be factored as U = ΦS, where S ∈ SO 2 (F), and Φ is a matrix encoding the determinant content of U : Together with the polar decomposition, we may then factor any A ∈ M * 2 (F) as where P A is Hermitian and positive semi-definite, Φ A is as defined in (15), and S A ∈ SO 2 (F).

Proposition 5. For
as det(S) = 1. Moreover, we know from Proposition 3 and the fact that S ∈ O 2 (F) that P A = P B . We note that under these conditions we must have Conversely, let det(A) = det(B) and Thus A ∼ B.
This proposition allows us to identify equivalence classes as As in the step from the GL quotient to the SL quotient, where the dense point "blew up" to F * , the step from O 2 to SO 2 "unfurls" each nonsingular class to an S F 's worth of points, encoding determinant information about the matrices in each class. Since the determinant represents the squared length of the corresponding spacetime vectors, we may regard the spheres as having radius equal to that shared squared length. We note that this interpretation is consistent with the limiting case of a light-like vector and the lack of sphere in that case (radius zero).
The separability properties, however, have little room to improve here:

Discussion of Results
In this article we analyzed the case of real and complex 2×2 nonzero matrices quotiented by the general and special linear groups, as well as the regular and special unitary/orthogonal groups. This was inspired by the insight from previous studies [2,16] that such spaces can encode relevant geometrical and physical information in their topologies. In [2], it is shown that the quotient P 0 GL (C) can be identified with projective 2-spinors at a spacetime point (the projective spinors forming a sphere). On the other hand, the projective line P 1 GL (C) is constructed by considering the free, rank 2 modules built over the ring of 2 × 2 matrices M 2 (C) modulo the right scalar action by GL 2 (C): where A, B, C, D ∈ M 2 (C) and Λ ∈ GL 2 (C). It is shown in [16] that P 1 GL (C) encodes information about Penrose's twistor correspondence [15]. In particular, P 1 GL (C) splits into two pieces, one which is homeomorphic to projective twistor space CP 3 , and another which is homeomorphic to complexified, compactified complex Minkowski spacetime CM c , and the non-Hausdorff topology of P 1 GL (C) facilitates the transfer of information between the two pieces according to the twistor correspondence. In the present article, the new spaces considered result in a non-Hausdorff combination of spheres and punctured lines in the D = SL case, a future-pointing light cone in the D = O case, and a product of spheres and a future-pointing light cone in the D = SO case.
Worth noting is that as the subgroup gets smaller, the separation properties improve (see Fig. 5). In the case of GL the result is a T 0 but not T 1 space (see Prop. 1), then in the case of SL the result is a T 1 but not T 2 space (see Prop. 2), and finally in the cases of both O and SO, the results are both T 2 (in fact, metrizable, see Props. 4 and 6). The separation properties improve as the subgroup gets smaller since we are identifying fewer elements, and we are able to trace this very clearly geometrically.
Moreover, when one denominator is a subgroup of another, we can understand how the quotients morph into each other by considering matrix factorizations. First consider the case SL 2 (F) ⊆ GL 2 (F). We may factor any matrix A as A = M Λ, (21) where M is a P 0 GL (F) representative for A, and Λ = a b c d ∈ GL 2 (F).
If A is nonsingular, we may factor further as (Here, sgn denotes the sign function.) Thus, if we can parameterize the set of all M (mod GL), then we have a parameterization for the nonsingular part of P 0 GL (F), whereas if we can parameterize the set of all M (mod GL) and the set of all Ω, then we have a parameterization of the nonsingular part of P 0 SL (F). The difference between the GL and SL cases thus lies in the Ω term, which is clearly parameterized by F − {0}. Of course, as we saw above, the nonsingular part in the P 0 GL (F) case is a single point, whereas the nonsingular part in the P 0 SL (F) case was shown to be F − {0}, as suggested by the factorization above. Thus, we can imagine that as we increase the denominator from SL to GL, the nonsingular part F − {0} gets identified into a point. On the other hand, for the singular case, as shown above, we can take Λ ∈ SL 2 in the factorization (21), so there is no need to factor further and introduce Ω. Topologically, this means that no additional identifications take place among the singular classes when passing from the SL quotient to a GL quotient.
Consider the case SO 2 (F) ⊆ O 2 (F). As we saw, we may factor a nonsingular matrix A as where Φ A encodes the determinant information of the orthogonal part of A and is parameterized by S 1 (F = C) or S 0 (F = R). Thus, for the nonsingular part of the quotient, the effect of passing from the SO quotient to the O