Lecture 1 Erlangen Programme: Preview

The simplest objects with non-commutative (but still associative) multiplication may be 2× 2 matrices with real entries. The subset of matrices of determinant one has the following properties:

• It is a closed set under multiplication since det(AB)=detA· detB.
• The identity matrix is in the set.
• Any such matrix has an inverse (since detA≠ 0).

In other words, these matrices form a group, the SL₂(ℝ) group [240]—one of the two most important Lie groups in analysis. The other group is the Heisenberg group [138]. By contrast, the ax+b group, which is used to build wavelets, is only a subgroup of SL₂(ℝ)—see the numerator in (1) below.

The simplest non-linear transforms of the real line—linear-fractional or Möbius maps—may also be associated with 2× 2 matrices, cf. [26]*Ch. 13 [262]*Ch. 3 [10]:

An enjoyable calculation shows that the composition of two transforms (1) with different matrices g₁ and g₂ is again a Möbius transform with a matrix equal the product g₁ g₂. In other words, (1) is a (left) action of SL₂(ℝ).

According to F. Klein’s Erlangen programme (which was influenced by S. Lie), any geometry deals with invariant properties under a certain transitive group action. For example, we may ask: What kinds of geometry are related to the SL₂(ℝ) action (1)?

The Erlangen programme has probably the highest rate of praised/actually used among mathematical theories, not only due to the large numerator but also due to the undeservedly small denominator. As we shall see below, Klein’s approach provides some surprising conclusions even for such over-studied objects as circles.

1.1 Make a Guess in Three Attempts

It is easy to see that the SL₂(ℝ) action (1) also makes sense as a map of complex numbers z=x+i y, i²=−1, assuming the denominator is non-zero. Moreover, if y>0, then g· z has a positive imaginary part as well, i.e. (1) defines a map from the upper half-plane to itself.

However, there is no need to be restricted to the traditional route of complex numbers only. Less-known double and dual numbers, see [339]*Suppl. C and Appendix B.1, also have the form z=x+ι y but with different assumptions for the imaginary unit ι : ι²=0 or ι²=1, respectively. We will write ε and є instead of ι within dual and double numbers, respectively. Although the arithmetic of dual and double numbers is different from that of complex numbers, e.g. they have divisors of zero, we are still able to define their transforms by (1) in most cases.

Three possible values −1, 0 and 1 of σ:=ι² will be referred to here as elliptic, parabolic and hyperbolic cases, respectively. We repeatedly meet such a division of various mathematical objects into three classes. They are named by the historic first example—the classification of conic sections—however the pattern persistently reproduces itself in many different areas: equations, quadratic forms, metrics, manifolds, operators, etc. We will abbreviate this separation as the EPH classification. The common origin of this fundamental division of any family with one parameter can be seen from the simple picture of a coordinate line split by zero into negative and positive half-axes:

Connections between different objects admitting EPH classification are not limited to this common source. There are many deep results linking, for example, the ellipticity of quadratic forms, metrics and operators, e.g. the Atiyah–Singer index theorem. On the other hand, there are still many white spots, empty cells, obscure gaps and missing connections between some subjects.

To understand the action (1) in all EPH cases we use the Iwasawa decomposition [240]*§ III.1 of SL₂(ℝ)=ANK into three one-dimensional subgroups A, N and K:

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Figure 1.1: Actions of the subgroups A and N by Möbius transformations. Transverse thin lines are images of the vertical axis, grey arrows show the derived action.

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Subgroups A and N act in (1) irrespective of the value of σ: A makes a dilation by α², i.e. z↦ α²z, and N shifts points to left by ν, i.e. z↦ z+ν. This is illustrated by Fig. 1.1.

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Figure 1.2: Action of the subgroup K. The corresponding orbits are circles, parabolas and hyperbolas shown by thick lines. Transverse thin lines are images of the vertical axis, grey arrows show the derived action.

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By contrast, the action of the third matrix from the subgroup K sharply depends on σ—see Fig. 1.2. In elliptic, parabolic and hyperbolic cases, K-orbits are circles, parabolas and (equilateral) hyperbolas, respectively. Thin traversal lines in Fig. 1.2 join points of orbits for the same values of φ and grey arrows represent “local velocities”—vector fields of derived representations.

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 (a)(b)

Figure 1.3: K-orbits as conic sections. Circles are sections by the plane EE′, parabolas are sections by PP′ and hyperbolas are sections by HH′. Points on the same generator of the cone correspond to the same value of φ.

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It is well known that any cycle is a conic section and an interesting observation is that corresponding K-orbits are, in fact, sections of the same two-sided right-angle cone, see Fig. 1.3. Moreover, each straight line generating the cone, see Fig. 1.3(b), is crossing corresponding EPH K-orbits at points with the same value of parameter φ from (3). In other words, all three types of orbits are generated by the rotations of this generator along the cone.

K-orbits are K-invariant in a trivial way. Moreover, since actions of both A and N for any σ are extremely “shape-preserving”, we find natural invariant objects of the Möbius map:

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Figure 1.4: Decomposition of an arbitrary Möbius transformation g into a product.

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Proof. We will show that, for a given g∈ SL₂(ℝ) and a cycle C, its image gC is again a cycle. Figure 1.4 provides an illustration with C being a circle, but our reasoning works in all EPH cases.

For a fixed C, there is the unique pair of transformations g′_n from the subgroup N and g′_a∈ A that the cycle g′_a g′_n C is exactly a K-orbit. This will be shown later in Exercise 7.

We make a decomposition of g (g′_a g′_n)⁻¹ into a product similar to (3):

Since g′_a g′_n C is a K-orbit, we have g_k (g′_a g′_n C)=g′_a g′_n C. Then:

All transformations from subgroups A and N preserve the shape of any cycles in an obvious way. Therefore, the last expression g_ag_n g′_a g′_n C represents a cycle and our proof is finished.

According to Erlangen ideology, we should now study the invariant properties of cycles.

1.2 Covariance of FSCc

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 (cycles in our case, functions in functional analysis, etc.) to be simply points of some larger space. This space should be equipped with an appropriate structure to hold information externally which previously described the inner properties of our objects.

A generic cycle is the set of points (u,v)∈ℝ² defined for all values of σ by the equation

This equation (and the corresponding cycle) is defined by a point (k, l, n, m) from a projective space ℙ³, since, for a scaling factor λ ≠ 0, the point (λ k, λ l, λ n, λ m) defines an equation equivalent to (4). We call ℙ³ the cycle space and refer to the initial ℝ² as the point space.

In order to obtain a connection with the Möbius action (1), we arrange numbers (k, l, n, m) into the matrix

with a new hypercomplex unit ι_c and an additional parameter s, usually equal to ± 1. The values of σ_c:=ι_c² are −1, 0 or 1 independent of the value of σ. The matrix (5) is the cornerstone of an extended Fillmore–Springer–Cnops construction (FSCc) [300, 65, 100].

The significance of FSCc in the Erlangen framework is provided by the following result:

There are several ways to prove (6). Either by a brute-force calculation (which can, fortunately, be performed by a CAS) in Section 4.3, or through the related orthogonality of cycles [65]—see the end of Section 1.4.

The important observation here is that our extended version of FSCc (5) uses an imaginary unit ι_c, which is not related to ι, that defines the appearance of cycles on the plane. In other words, any EPH type of geometry in the cycle space ℙ³ admits drawing of cycles in the point space ℝ² as circles, parabolas or hyperbolas. We may think of points of ℙ³ as ideal cycles while their depictions on ℝ² are only their shadows on the wall of Plato’s cave.

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(a)  (b)

Figure 1.5: Cycle implementations, centres and foci. (a) Different EPH implementations of the same cycles defined by quadruples of numbers. (b) Centres and foci of two parabolas with the same focal length.

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Figure 1.5(a) shows the same cycles drawn in different EPH styles. We note the first order contact between the circle, parabola and hyperbola in the intersection points with the real line. Informally, we can say that EPH realisations of a cycle look the same in a vicinity of the real line. It is not surprising since cycles are invariants of the hypercomplex Möbius transformations, which are extensions of SL₂(ℝ)-action (4) on the real line.

Points c_e,p,h=(l/k, −σ_c n/k) on Fig. 1.5(a) are the respective e/p/h-centres of drawn cycles. They are related to each other through several identities:

Figure 1.5(b) presents two cycles drawn as parabolas. They have the same focal length, n/2k, and, thus, their e-centres are on the same level. In other words, concentric parabolas are obtained by a vertical shift, not by scaling, as an analogy with circles or hyperbolas may suggest.

Figure 1.5(b) also presents points, called e/p/h-foci,

which are independent of the sign of s. If a cycle is depicted as a parabola, then the h-focus, p-focus and e-focus are, correspondingly, the geometrical focus of the parabola, its vertex, and the point on the directrix nearest to the vertex.

As we will see (cf. Theorems 5 and 7), all three centres and three foci are useful attributes of a cycle, even if it is drawn as a circle.

1.3 Invariants: Algebraic and Geometric

We use known algebraic invariants of matrices to build appropriate geometric invariants of cycles. It is yet another demonstration that any division of mathematics into subjects is only illusive.

For 2× 2 matrices (and, therefore, cycles), there are only two essentially different invariants under similarity (6) (and, therefore, under Möbius action (1)): the trace and the determinant. The latter was already used in (8) to define a cycle’s foci. However, due to the projective nature of the cycle space ℙ³, the absolute values of the trace or determinant are irrelevant, unless they are zero.

Alternatively, we may have a special arrangement for the normalisation of quadruples (k,l,n,m). For example, if k≠0, we may normalise the quadruple to (1,l/k,n/k,m/k) with the highlighted cycle’s centre. Moreover, in this case, −detC_σc^s is equal to the square of cycle’s radius, cf. Section 1.6. Another normalisation, detC_σc^s=±1, is used in [163] to obtain a good condition for touching circles.

We still have important characterisation even with non-normalised cycles. For example, invariant classes (for different σ_c) of cycles are defined by the condition detC_σc^s=0. Such a class is parameterised only by two real numbers and, as such, is easily attached to a certain point of ℝ². For example, the cycle C_σc^s with detC_σc^s=0, σ_c=−1, drawn elliptically, represents just a point (l/k,n/k), i.e. an (elliptic) zero-radius circle. The same condition with σ_c=1 in hyperbolic drawing produces a null-cone originating at point (l/k,n/k):

i.e. a zero-radius cycle in a hyperbolic metric.

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Figure 1.6: Different σ-implementations of the same σc-zero-radius cycles. The corresponding foci belong to the real axis.

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In general, for every concept there are at least nine possibilities: three EPH cases in the cycle space times three EPH realisations in the point space. These nine cases for “zero-radius” cycles are shown in Fig. 1.6. For example, p-zero-radius cycles in any implementation touch the real axis, that is they are horocycles.

This “touching” property is a manifestation of the boundary effect in the upper-half plane geometry. The famous question about hearing a drum’s shape has a counterpart: Can we see/feel the boundary from inside a domain?

Both orthogonality relations described below are “boundary-aware” as well. After all, it is not surprising since SL₂(ℝ) action on the upper half-plane was obtained as an extension of its action (1) on the boundary.

According to the categorical viewpoint, internal properties of objects are of minor importance in comparison to their relations with other objects from the same class. As an example, we may give the proof of Theorem 3 described at the end of of the next section. Thus, from now on, we will look for invariant relations between two or more cycles.

1.4 Joint Invariants: Orthogonality

The most expected relation between cycles is based on the following Möbius-­invariant “inner product”, built from a trace of the product of two cycles as matrices:

Here, S _σc^s means the complex conjugation of elements of the matrix S _σc^s. Notably, an inner product of this type is used, for example, in GNS construction to make a Hilbert space out of C^*-algebra. The next standard move is given by the following definition:

For the case of σ_c σ=1, i.e. when the geometries of the cycle and point spaces are both either elliptic or hyperbolic, such an orthogonality is standard, defined in terms of angles between tangent lines in the intersection points of two cycles. However, in the remaining seven (9−2) cases, the innocent-looking Definition 4 brings unexpected relations.

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(a)(b)(c)

Figure 1.7: Orthogonality of the first kind in the elliptic point space. Each picture presents two groups (green and blue) of cycles which are orthogonal to the red cycle Cσcs. Point b belongs to Cσcs and the family of blue cycles passing through b is orthogonal to Cσcs. They all also intersect at the point d which is the inverse of b in Cσcs. Any orthogonality is reduced to the usual orthogonality with a new (“ghost”) cycle (shown by the dashed line), which may or may not coincide with Cσcs. For any point a on the “ghost” cycle, the orthogonality is reduced to the local notion in terms of tangent lines at the intersection point. Consequently, such a point a is always the inverse of itself.

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Elliptic (in the point space) realisations of Definition 4, i.e. σ=−1, are shown in Fig. 1.7. Figure 1.7(a) corresponds to the elliptic cycle space, e.g. σ_c=−1. The orthogonality between the red circle and any circle from the blue or green families is given in the usual Euclidean sense. Figure 1.7(b) (parabolic in the cycle space) and Fig. 1.7(c) (hyperbolic) show the non-local nature of the orthogonality. There are analogous pictures in parabolic and hyperbolic point spaces—see Section 6.1.

This orthogonality may still be expressed in the traditional sense if we will associate to the red circle the corresponding “ghost” circle, which is shown by the dashed lines in Fig. 1.7. To describe the ghost cycle, we need the Heaviside function χ(σ):

The above connection between various centres of cycles illustrates their relevance to our approach.

One can easily check the following orthogonality properties of the zero-radius cycles defined in the previous section:

1. Due to the identity ⟨ C_σc^s,C_σc^s ⟩=2det C_σc^s, zero-radius cycles are self-ortho­go­nal (isotropic).
2. A cycle C_σc^s is σ-orthogonal to a zero-radius cycle Z_σc^s if and only if C_σc^s passes through the σ-centre of Z_σc^s.

Proof.[Sketch of proof of Theorem 3] The validity of Theorem 3 for a zero-radius cycle

with the centre z=x+ι y is straightforward. This implies the result for a generic cycle with the help of Möbius invariance of the product (9) (and, thus, the orthogonality) and the above relation (Theorem 2) between the orthogonality and the incidence. See Exercise 7 for details.

1.5 Higher-order Joint Invariants: Focal Orthogonality

With our appetite already whet we may wish to build more joint invariants. Indeed, for any polynomial p(x₁,x₂,…,x_n) of several non-commuting variables, one may define an invariant joint disposition of n cycles ^jC_σc^s by the condition

However, it is preferable to keep some geometrical meaning of constructed notions.

An interesting observation is that, in the matrix similarity of cycles (6), one may replace element g∈SL₂(ℝ) by an arbitrary matrix corresponding to another cycle. More precisely, the product C_σc^sS _σc^sC_σc^s is again the matrix of the form (5) and, thus, may be associated with a cycle. This cycle may be considered as the reflection of S _σc^s in C_σc^s.

Due to invariance of all components in the above definition, f-orthogonality is a Möbius-invariant condition. Clearly, this is not a symmetric relation—if C_σc^s is f-orthogonal to S _σc^s, then S _σc^s is not necessarily f-orthogonal to C_σc^s.

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Figure 1.8: Focal orthogonality for circles. To highlight both similarities and distinctions with ordinary orthogonality, we use the same notations as in Fig. 1.7.

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Figure 1.8 illustrates f-orthogonality in the elliptic point space. In contrast to Fig. 1.7, it is not a local notion at the intersection points of cycles for all σ_c. However, it may again be clarified in terms of the appropriate f-ghost cycle, cf. Theorem 5.

Note the above intriguing interplay between the cycle’s centres and foci. Although f-orthogonality may look exotic, it will appear again at the end of next section.

Of course, it is possible to define other interesting higher-order joint invariants of two or even more cycles.

1.6 Distance, Length and Perpendicularity

Geometry in the plain meaning of the word deals with distances and lengths. Can we obtain them from cycles?

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(a)   (b)   

Figure 1.9: Radius and distance for parabolas. (a) The square of the parabolic diameter is the square of the distance between roots if they are real (z1 and z2). Otherwise, it is the negative square of the distance between the adjoint roots (z3 and z4). (b) Distance is the extremum of diameters in elliptic (z1 and z2) and parabolic (z3 and z4) cases.

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We already mentioned that, for circles normalised by the condition k=1, the value −det C_σc^s=−1/2⟨ C_σc^s,C_σc^s ⟩ produces the square of the traditional circle radius. Thus, we may keep it as the definition of the σ_c-radius for any cycle. However, we then need to accept that, in the parabolic case, the radius is the (Euclidean) distance between (real) roots of the parabola—see Fig. 1.9(a).

Having already defined the radii of circles, we may use them for other measurements in several different ways. For example, the following variational definition may be used:

If σ_c=σ, this definition gives, in all EPH cases, the following expression for a distance d_e,p,h(u,v) between the endpoints of any vector w=u+i v:

The parabolic distance d_p²=u², see Fig. 1.9(b), sits algebraically between d_e and d_h according to the general principle (2) and is widely accepted [339]. However, one may be unsatisfied by its degeneracy.

An alternative measurement is motivated by the fact that a circle is the set of equidistant points from its centre. However, there are now several choices for the “centre”: it may be either a point from three centres (7) or three foci (8).

This definition is less common and has some unusual properties like non-symmetry: l_f(AB)≠ l_f(BA). However, it comfortably fits the Erlangen programme due to its SL₂(ℝ)-conformal invariance:

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Figure 1.10: The perpendicular as the shortest route to a line.

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We may return from lengths to angles noting that, in the Euclidean space, a perpendicular is the shortest route from a point to a line—see Fig. 1.10.

A pleasant surprise is that l_f-perpendicularity obtained using the length from focus (Definition 9) coincides with f-orthogonality (already defined in Section 1.5), as follows from Theorem 1.

All these study are waiting to be generalised to higher-dimensions. Quaternions and Clifford algebras provide a suitable language for this [191, 273].

1.7 The Erlangen Programme at Large

As we already mentioned, the division of mathematics into separate areas is purely imaginary. Therefore, it is unnatural to limit the Erlangen programme only to “geometry”. We may continue to look for SL₂(ℝ)-invariant objects in other related fields. For example, transform (1) generates the unitary representation on the space L₂(ℝ):

The above transformations have two invariant subspaces within L₂(ℝ): the Hardy space and its orthogonal complements. These spaces are of enormous importance in harmonic analysis. Similar transformations

for m=2, 3, …, can be defined on square-integrable functions in the upper half-plane. The respective invariant subspaces are weighted Bergman spaces of complex-valued analytic and poly-analytic functions.

Transformations (14) produce the discrete series representations of SL₂(ℝ), cf. [240]*§ IX.2. Consequently, all main objects of complex analysis can be obtained in terms of invariants of these representations. For example:

• The Cauchy and Bergman integrals are special cases of wavelet transform, cf. [176]*§ 3. The corresponding mother wavelets are eigenfunctions of operators from the representations of the subgroup K.
• The Cauchy–Riemann and Laplace operators are invariant differential operators. They annihilate the respective mother wavelet which are mentioned above, cf. [193]*§ 3.
• Taylor series is a decomposition over eigenfunctions of the subgroup K, cf. [170]*§ 3.4.

It would be an omission to limit this construction only to the discrete series, complex numbers and the subgroup K. Two other series of representations (principal and complimentary—see [240]*§ VI.6) are related to important special functions [333] and differential operators [254]. These series have unitary realisations in dual and double numbers [170, 194]. Their relations with subgroups N′ and will be shown in Section 3.3.4.

Moving further, we may observe that transform (1) is also defined for an element x in any algebra A with a unit 1 as soon as (cx+d1)∈A has an inverse. If A is equipped with a topology, e.g. is a Banach algebra, then we may study a functional calculus for element x [168] in this way. It is defined as an intertwining operator between the representations (13) or (14) in spaces of analytic functions and similar representations in left A-modules.

In the spirit of the Erlangen programme, such functional calculus is still a geometry, since it is dealing with invariant properties under a group action. However, even for the simplest non-normal operator, e.g. a Jordan block of length k, the obtained space is not like a space of points but is, rather, a space of k-th jets [182]. Such non-point behaviour is often attributed to non-commutative geometry and the Erlangen programme provides an important insight on this popular topic [176].

Of course, there is no reason to limit the Erlangen programme to the group SL₂(ℝ) only—other groups may be more suitable in different situations. For example, the Heisenberg group and its hypercomplex representations are useful in quantum mechanics [199, 196]. However, SL₂(ℝ) still possesses large unexplored potential and is a good object to start with.