Lecture 4 The Extended Fillmore–Springer–Cnops Construction

Cycles—circles, parabolas and hyperbolas—are invariant families under their respective Möbius transformations. We will now proceed with a study of the invariant properties of cycles according to the Erlangen programme. A crucial step is to treat cycles not as subsets of ℝⁿ but rather as points of some projective space of higher dimensionality, see [33]*Ch. 3 [59] [276]. It is common in algebra to represent quadratic forms by square matrices, thus circles and their Möbius transformations are naturally described by 2× 2 matrices, see [306]*Ch. 9 [300]*Ch. 1. Yet one important component is still missing in those works: the invariant cycle product. This enhancement was independently introduced by several researchers in hypercomplex context [65, 100], see also [163]. This chapter introduces a further extension: the signature of the cycle project can vary even for the same type of point space.

4.1 Invariance of Cycles

K-orbits, shown in Fig. 1.2, are K-invariant in a trivial way. Moreover, since actions of both A and N for any σ are extremely ‘shape-preserving’, see Exercise 12, we meet natural invariant objects of the Möbius map:

It is known, from analytic geometry, that a cycle is defined by the equation

where σ = ι² and k, l, n, m are real parameters, such that not all of them are equal to zero. Using hypercomplex numbers, we can write the same equation as, cf. [339]*Suppl. C(42a),

where w= u+ι v, K=ι k, L=n+ι l, M=ι m and conjugation is defined by w= u−ι v.

All three EPH types of cycles enjoy many common properties, sometimes even beyond those which we normally expect. For example, the following definition is quite intelligible even when extended from the above elliptic and hyperbolic cases to the parabolic one:

The usefulness of Defn. 3, even in the parabolic case, will be justified, for example, by

• the uniformity of the description of relations between the centres of orthogonal cycles, see the next subsection and Fig. 6.3,
• the appearance of concentric parabolas in Fig. 10.3(P_e:N and P_h:N).

Here is one more example of the natural appearance of concentric parabolas:

This property is classical for circles [71]*§ 2.3 and is also known for parabolas [339]*§ 10. However, for parabolas, Yaglom used the word ‘concentric’ in quotes, since he did not define the centres of a parabola explicitly.

The family of all cycles from Definition 1 is invariant under Möbius transformations (4) in all EPH cases, which was already stated in Theorem 2. If a cycle does not intersect the real axis then the proof of Theorem 2 has the only gap: a demonstration that we can always transform the cycle to a K-orbit.

To finish the proof of Theorem 2 we need to consider two more cases which are also related to the structure of the group SL₂(ℝ):

If all the above details are taking into consideration the proof of Theorem 2 may not appear as simple as we may initially thought. However, it still highlights important connections between the invariance of cycles and the structure of the group SL₂(ℝ). Furthermore, we fully describe how cycles are transformed by Möbius transformations in Theorem 13.

4.2 Projective Spaces of Cycles

Figure 1.3 suggests that we may obtain a unified treatment of cycles in all EPH cases by consideration of higher-dimension spaces. The standard mathematical method is to declare objects under investigation to be simply points of some larger space.

If an object is considered as a point (in a new space) all information about its inner structure is lost, of course. Therefore, this space should be equipped with an appropriate structure to hold information externally which previously described the inner properties of our objects. In other words, the inner structure of an object is now revealed through its relations to its peers¹.

In the geometry, the similar ideas are known as representational geometry, see [101, 99] and references there for a discussion. We will utilise the this fundamental approach for cycles. A generic cycle from Definition 1 is the set of points (u,v)∈ℝ_σ defined for the respective values of σ by the equation

This equation (and the corresponding cycle) is completely determined by a point (k, l, n, m)∈ ℝ⁴. However, this is not a one-to-one correspondence. For a scaling factor λ ≠ 0, the point (λ k, λ l, λ n, λ m) defines an equivalent equation to (3). Thus, we prefer to consider the projective space ℙ³, that is, ℝ⁴ factorised by the equivalence relation (k, l, n, m)∼ (λ k, λ l, λ n, λ m) for any real λ ≠ 0. A good introductory read on projective spaces is [296]*Ch. 10, see also Ch. 13 in [311]. The chapter in the later book is titled “Projective Geometry Is All Geometry”, which is a quote from Arthur Cayley and our consideration below is a further illustration to this claim.

We also note that the Equation (2) of a cycle can be written as a quadratic form

in the homogeneous coordinates (w₁,w₂), such that w=w₁/w₂. Since quadratic forms are related to square matrices, see Section 4.4, we define another map on the cycle space as follows.

The matrix (5) is the cornerstone of the (extended) Fillmore–Springer–Cnops construction (FSCc) [65, 100] and is closely related to the technique recently used by A.A. Kirillov to study the Apollonian gasket [163]. A hint for the composition of this matrix is provided by the following exercise.

Identifications of both Q and M are straightforward. Indeed, a point (k, l, n, m)∈ℙ³ equally well represents (as soon as σ , σ_c and s are already fixed) both the Equation (3) and the line of matrix (5). Thus, for fixed σ , σ_c and s, one can introduce the correspondence between quadrics and matrices shown by the horizontal arrow on the following diagram:

which combines Q and M.

4.3 Covariance of FSCc

At first glance, the horizontal arrow in (6) seems to be of little practical interest since it depends on too many different parameters (σ, σ_c and s). However, the following result demonstrates that it is compatible with simple calculations of cycles’ images under the Möbius transformations.

Proof. There are several ways to prove (7), but for now we present a brute-­force calculation, which can fortunately be performed by a CAS [191]. See Appendix C for information on how to install (Section C.2) and start (Section C.3) the software, which will be sufficient for this proof. Further use of the software would be helped by learning which methods are available from the library (Section C.4) and which predefined objects exist upon initialisation (Section C.5). Assuming the above basics are known we proceed as follows.

First, we build a cycle passing a given point P=[u, v]. For this, a generic cycle C with parameters (k,l,n,m) is bounded by the corresponding condition:

In [2]: C2=C.subject_to(C.passing(P))

Then, we build the conjugated cycle with a generic g= (

)∈SL₂(ℝ) and a hypercomplex unit es and parameter s=±1:

In [3]: C3=C2.sl2_similarity(a, b, c, d, es, matrix([[1,0],[0,s]])

We also find the image of P under the Möbius transformation with the same element of g∈SL₂(ℝ) but a different hypercomplex unit e:

In [4]: P1=clifford_moebius_map(sl2_clifford(a, b, c, d, e), P, e)

Finally, we check that the conjugated cycle C3 passes the Möbius transform P1. A simplification based on the determinant value 1 and s=±1 will be helpful:

In [5]: print  C3.val(P1).subs([a==(1+b*c)/d,pow(s,2)==1]) \
                .normal().is_zero() 
Out[5]: True

Thus, we have confirmation that the theorem is true in the stated generality. One may wish that every mathematical calculation can be done as simply.

The above proof cannot satisfy everyone’s aesthetic feeling. For this reason, an alternative route based on orthogonality of cycles [65] will be given later—see Exercise 7.

It is worth noticing that the image S _σc^s under similarity (7) is independent of the values of s and σ_c. This, in particular, follows from the following exercise.

Now we have an efficient tool for investigating the properties of some notable cycles, which have appeared before.

These easy examples also show that the software is working as expected.

4.4 Origins of FSCc

The Fillmore–Springer–Cnops construction, in its generalised form, will play a central rôle in our subsequent investigation. Thus, it is worth looking at its roots and its origins before we begin using it. So far, it has appeared from the thin air, but can we intentionally invent it? Are there further useful generalisations of FSCc? All these are important questions and we will make an attempt to approach them here.

As is implied by its name, FSCc was developed in stages. Moreover, it appeared independently in a different form in the recent work of Kirillov [163]. This indicates the naturalness and objectivity of the construction. We are interested, for now, in the flow of ideas rather than exact history or proper credits. For the latter, the reader may consult the original works [100] [64] [65] [163] [289]*Ch. 18 as well as references therein. Here, we treat the simplest two-dimensional case. In higher dimensions, non-commutative Clifford algebras are helpful with some specific adjustments.

4.4.1 Projective Coordiantes and Polynomials

An important old observation is that Möbius maps appear from linear transformations of homogeneous (projective) coordinates, see [269]*Ch. 1 for this in a context of invariant theory. This leads to the FSCc in several steps:

1. Take a real projective space ℙ¹ as a quotient of ℝ² by the equivalence relation (x,y)∼ (tx,ty) for t≠ 0. Then, any line with y≠ 0 can be represented by (x/y,1). Thus, the invertible linear transformation

   ⎛ ⎜ ⎝       a b c d ⎞ ⎟ ⎠

   ⎛ ⎜ ⎝       x y ⎞ ⎟ ⎠

   =

   ⎛ ⎜ ⎝       a x+b y c x+d y ⎞ ⎟ ⎠

   ,   g=

   ⎛ ⎜ ⎝       a b c d ⎞ ⎟ ⎠

   ∈SL2(ℝ),

   ⎛ ⎜ ⎝       x y ⎞ ⎟ ⎠

   ∈ℝ2 (9)

   will become the Möbius transformation (3) on the representatives (

   u

   1

   ). Similarly, we can consider Möbius actions on complex numbers w=u+i v.
2. The next observation [64] is that, if we replace the vector (

   w

   1

   ), w∈ℂ by a 2× 2 matrix (

   w	w
   1	1

   ), then the matrix multiplication with g∈SL₂(ℝ) will transform it as two separate copies of the vector in (9). The “twisted square” of this matrix is

   Z=

   ⎛ ⎜ ⎝       w −ww 1 −w ⎞ ⎟ ⎠

   =

   1 2

   ⎛ ⎜ ⎝       w w 1 1 ⎞ ⎟ ⎠

   ⎛ ⎜ ⎝       1 −w 1 −w ⎞ ⎟ ⎠

   . (10)

   Then, the linear action (9) on vectors is equivalent to the similarity g Z g⁻¹ for the respective matrix Z from (10).
3. Finally, one can link matrices Z in (10) with zero-radius circles, see Exercise 1, which are also in one-to-one correspondences with their centres. Then, the above similarity Z↦ g Z g⁻¹ can be generalised to the action (7).

Another route was used in a later book of Cnops [65]: a predefined geometry of spheres, specifically their orthogonality, was encoded in the respective matrices of the type (5). Similar connections between geometry of cycles and matrices lead Kirillov, see [163]*§ 6.3 and the end of this section. He arrived at an identification of disks with Hermitian matrices (which is similar to FSCc) through the geometry of Minkowski space-time and the intertwining property of the actions of SL₂(ℂ).

There is one more derivation of FSCc based on projective coordinates. We can observe that the homogeneous form (4) of cycle’s equation (2) can be written using matrices as follows:

Then, the linear action (9) on vectors (

) corresponds to conjugated action on 2× 2 matrices (

) by the intertwining identity:

where the respective actions of SL₂(ℝ) on vectors and matrices are

In other words, we obtained the usual FSCc with the intertwining property of the type (7). However, the generalised form FSCc does not result from this consideration yet.

Alternatively, we can represent the Equation (4) as:

Then, the intertwining relation similar to (12) holds if matrix similarity is replaced by the matrix congruence:

This identity provides a background to the Kirillov correspondence between circles and matrices, see [163]*§ 6.3 and [306]*§ 9.2. Clearly, it is essentially equivalent to FSCc and either of them may be depending on a convenience. It does not hint at the generalised form of FSCc either.

We will mainly work with the FSCc, occasionally stating equivalent forms of our results for the Kirillov correspondence.

4.4.2 Co-Adjoint Representation

In the above construction, the identity (11) requires the same imaginary unit to be used in the quadratic form (the left-hand side) and the FSCc matrix (the right-hand side). How can we arrive at the generalised FSCc directly without an intermediate step of the standard FSCc with the same type of EPH geometry used in the point cycle spaces? We will consider a route originating from the representation theory.

Any group G acts on itself by the conjugation g: x ↦ g x g⁻¹, see 1. This map obviously fixes the group identity e. For a Lie group G, the tangent space at e can be identified with its Lie algebra g, see Subsection 2.3.1. Then, the derived map for the conjugation at e will be a linear map g → g. This is an adjoint representation of a Lie group G on its Lie algebra. This is the departure point for Kirillov’s orbit method, which is closely connected to induced representations, see [159, 161].

We can also note that the above transformation fixes matrices (

) which are scalar multiples of the identity matrix. Thus, we can consider the conjugated action (13) of SL₂(ℝ) on the pairs of matrices, or intervals in the matrix space of the type

In other words, we obtained the generalised FSCc, cf. (5), and the respective action of SL₂(ℝ). Furthermore, a connection of the above pairs of matrices with the cycles on a plane can be realised through the Poincaré extension, see Ch. 14.

4.5 Projective Cross Ratio

An important invariant of Möbius transformations in complex numbers is the cross ratio of four distinct points, see [26]*§ 13.4:

Zero divisors make it difficult to handle a similar fraction in dual and double numbers. Projective coordinates are again helpful in this situation—see [52]. Define the symplectic form, cf. [11]*§ 41, of two vectors by

The one-dimensional projective space ℙ¹(ℝ_σ) is the quotient of ℝ_σ² with respect to the equivalence relation: z∼ z′ if their symplectic form vanishes. We also say [52] that two points in ℙ¹(ℝ_σ) are essentially distinct if there exist their representatives z, z′∈ℝ_σ² such that ω(z,z′) is not a zero divisor.

We say that a collection of points of ℙ¹(ℝ_σ) is concyclic if all their representatives in ℝ_σ² satisfy to Equation (11) for some FSCc matrix (

) .

We have seen another way to obtain FSCc—from the projective cross ratio.

1

The same is true for human beings.