Lecture 3 Homogeneous Spaces from the Group SL₂(ℝ)

Now we adopt the previous theoretical constructions for the particular case of the group SL₂(ℝ). It is common to present SL₂(ℝ) solely as the transformation group of the Lobachevsky half-plane. We take a wider viewpoint considering SL₂(ℝ) as an abstract group and wish to classify all its realizations as a transformation group of certain sets. More specifically, we describe all homogeneous spaces SL₂(ℝ)/H, where H is a closed continuous subgroup of SL₂(ℝ), see Section 2.2.2. To begin with, we start from the two-dimensional subgroup.

3.1 The Affine Group and the Real Line

The affine group of the real line, also known as the ax+b group , can be identified with either subgroup of lower- or upper-triangular matrices of the form:

These subgroups are obviously conjugates of each other and we can consider only the subgroup F here. We are following the construction from Section 2.2.2 and using its notations.

Firstly, we address the simpler case of the subgroup F′ of all lower triangular matrices in SL₂(ℝ), which admits both positive and negative entries on the main diagonal. The corresponding homogeneous space X=SL₂(ℝ)/F′ is one-di­men­sio­nal and can be parametrised by real numbers. We define the natural projection p as:

since there is the following decomposition: 1

The decomposition shows that each coset in SL₂(ℝ)/F′ either

• contains exactly one upper-triangular matrix of the form (

  1	u
  0	1

  ) and, thus, can be parametrised by the real number u; or
• is the coset of matrices (

  a	b
  −b−1	0

  ) which represents ∞.

Then, we define the smooth map s to be a right inverse of p:

The corresponding map r(g)=s(p(g))⁻¹g is calculated to be:

Consequently, the decomposition g=s(p(g))r(g) is given by (a).

Therefore the action of SL₂(ℝ) on SL₂(ℝ)/F′ is the Möbius (linear-fractional) transformations of the projective real line $ℝ^_$=ℝ∪{∞}:

In Section 4.4, we will see the relevance of this action to projective spaces. 4

To describe the homogeneous space SL₂(ℝ)/F and SL₂(ℝ)-action on it, we rewrite the identity (a) for d≠ 0 as:

where χ is the Heaviside function (10). The last matrix in the right-hand side of (b) belongs to F and is uniquely defined by this identity. Thus, cosets representing matrices with d≠ 0 are parametrised by ℝ×{−1,+1}. The respective realisations of maps from Section 2.2.2 are:

Thus, the homogeneous space SL₂(ℝ)/F can be identified with the double cover of the projective real line. SL₂(ℝ)-action on SL₂(ℝ)/F in terms of the above maps (c)–(e) is:

Here is some properties of the Möbius transformations of the real line:

3.2 One-dimensional Subgroups of SL₂(ℝ)

Any element X of the Lie algebra sl₂ defines a one-parameter continuous subgroup g_X(t)=e^tX, t∈ℝ of SL₂(ℝ). Two such subgroups g_X(t) and g_Y(t) with X=aY for a non-zero a∈ ℝ coincide as sets and we can easily re-parameterise them: g_Y(t)=g_X(at). Therefore we will not distinguish such subgroups in the following.

We already listed four one-parameter continuous subgroups in Exercise 3 and can provide further examples, e.g. the subgroup of lower-triangular matrices. However, there are only three different types of subgroups under the matrix similarity A↦ MAM⁻¹.

Proof. Any one-parameter subgroup is obtained through the exponential map, see Section 2.3,

of an element X of the Lie algebra sl₂ of SL₂(ℝ). Such an X is a 2× 2 matrix with zero trace. The behaviour of the Taylor expansion (4) depends on the properties of the powers Xⁿ. This can be classified by a straightforward calculation:

It is a simple exercise in characteristic polynomials to see that, through the matrix similarity, we can obtain from X a generator

• of the subgroup K if (−detX) <0,
• of the subgroup N if (−detX) =0,
• of the subgroup A if (−detX) >0.

The determinant is invariant under the similarity. Thus, these cases are distinct.

We will often use subgroups and N′ as representatives of the corresponding equivalence classes under matrix conjugation.

An interesting property of the subgroups A, N and K is their appearance in the Iwasawa decomposition [240]*§ III.1 of SL₂(ℝ)=ANK in the following sense. Any element of SL₂(ℝ) can be represented as the product:

The Iwasawa decomposition shows once more that SL₂(ℝ) is a three-dimensional manifold. A similar decomposition G=ANK is possible for any semisimple Lie group G, where K is the maximal compact group, N is nilpotent and A normalises N. Although the Iwasawa decomposition will be used here on several occasions, it does not play a crucial role in the present consideration. Instead, Proposition 2 will be the cornerstone of our construction.

3.3 Two-dimensional Homogeneous Spaces

Here, we calculate the action of SL₂(ℝ) (3) (see Section 2.2.2) on X=SL₂(ℝ)/H for all three possible one-dimensional subgroups H=, N′ or K. Counting dimensions (3−1=2) suggests that the corresponding homogeneous spaces are two-dimensional manifolds. In fact, we identify X in each case with a subset of ℝ² as follows. First, for every equivalence class of SL₂(ℝ)/H we chose a representative, which is an upper-triangular matrix.

The existence of such a triangular matrix will be demonstrated in each case separately. Now, we define the projection p:SL₂(ℝ)→ X, assigning p(g)=(a₁b₁,a₁²), where (

) is the upper-triangular matrix¹ representing the equivalence class of g. We also choose [170]*p. 108 the map s: X→ G in the form:

This formula will be used for all three possible subgroups H.

3.3.1 From the Subgroup K

The homogeneous space SL₂(ℝ)/K is the most traditional case in representation theory. The maps p and s defined above produce the following decomposition g=s(p(g))r(g):

Then, the SL₂(ℝ)-action defined by the formula g· x=p(g*s(x)) (3) takes the form:

Obviously, it preserves the upper half-plane v>0. The expression (10) is very cumbersome and it can be simplified by the complex imaginary unit i²=−1, which reduces (10) to the Möbius transformation

We need to assign a meaning to the case cw+d=0 and this can be done by the addition of an infinite point ∞ to the set of complex numbers—see, for example, [26]*Definition 13.1.3 for details.

In this case, complex numbers appeared naturally.

3.3.2 From the Subgroup N′

We consider the subgroup of lower-triangular matrices N′ (6). For this subgroup, the representative of cosets among the upper-triangular matrices will be different. Therefore, we receive an apparently different decomposition g=s(p(g))r(g), cf. (9)

We postpone the treatment of the exceptional case d=0 until Section 8.1. The SL₂(ℝ)-action (3) now takes the form:

This map preserves the upper half-plane v>0 just like the case of the subgroup K. The expression (13) is simpler than (10), yet we can again rewrite it as a linear-fractional transformation with the help of the dual numbers unit ε ²=0:

We briefly review the algebra of dual numbers in Appendix B.1. Since they have zero divisors, the fraction is not properly defined for all cu+d=0. The proper treatment will be considered in Section 8.1 since it is not as simple as in the case of complex numbers.

3.3.3 From the Subgroup

In the last case of the subgroup , we still can obtain the decomposition

However, the new aspect is here that (15) presents the decomposition g=s(p(g))r(g) if and only if d²−c²>0. Otherwise the matrix (

) is not in . If d²−c²<0 we use the decomposition 15

The modified maps p(g): G → G/H is:

The respective modification of the map s: G/H → G is:

The geometrical meaning of this modification is that the homogeneous spaces G/K and G/N are parametrised by points of the upper half-plane, while G/ shall be parametrised by the union of the upper and lower half-planes. We will further discuss this difference Section 8.2. The exceptional situation d=± c will be treated in Section 8.1.

The SL₂(ℝ)-action (3) takes the form

Notably, this time the map does not preserve the upper half-plane v>0: the sign of ( c u+d)² −(cv)² is not determined. To express this map as a Möbius transformation, we require the double numbers (also known as split-complex numbers) unit є ²=1:

The algebra of double numbers is briefly introduced in Appendix B.1.

3.3.4 Unifying All Three Cases

There is an obvious similarity in the formulae obtained in each of the above cases. To present them in a unified way, we introduce the parameter σ which is equal to −1, 0 or 1 for the subgroups K, N′ or , respectively. Then, decompositions (9), (12) and (15) are

The respective SL₂(ℝ)-actions on the homogeneous space SL₂(ℝ)/H, where H=, N′ or K, are given by

Finally, this action becomes the linear-fractional (Möbius) transformation for hypercomplex numbers in two-dimensional commutative associative algebra (see Appendix B.1) spanned by 1 and ι:

Thus, a comprehensive study of SL₂(ℝ)-homogeneous spaces naturally introduces three number systems. Obviously, only one case (complex numbers) belongs to mainstream mathematics. We start to discover empty cells in our periodic table.

3.4 Elliptic, Parabolic and Hyperbolic Cases

As we have seen in the previous section, there is no need to be restricted to the traditional route of complex numbers only. The arithmetic of dual and double numbers is different from complex numbers mainly in the following aspects:

1. They have zero divisors. However, we are still able to define their transforms by (4) in most cases. The proper treatment of zero divisors will be done through corresponding compactification—see Section 8.1.
2. They are not algebraically closed. However, this property of complex numbers is not used very often in analysis.

We have agreed in Section 1.1 that three possible values −1, 0 and 1 of σ:=ι² will be referred to here as elliptic, parabolic and hyperbolic cases, respectively. This separation into three cases will be referred to as the EPH classification. Unfortunately, there is a clash here with the already established label for the Lobachevsky geometry. It is often called hyperbolic geometry because it can be realised as a Riemann geometry on a two-sheet hyperboloid. However, within our framework, the Lobachevsky geometry should be called elliptic and it will have a true hyperbolic counterpart.

3.5 Orbits of the Subgroup Actions

We start our investigation of the Möbius transformations (4)

on the hypercomplex numbers w=u+ι v from a description of orbits produced by the subgroups A, N and K. Due to the Iwasawa decomposition (7), any Möbius transformation can be represented as a superposition of these three actions.

The actions of subgroups A and N for any kind of hypercomplex numbers on the plane are the same as on the real line: A dilates and N shifts—see Fig. 1.1 for illustrations. Thin traversal lines in Fig. 1.1 join points of orbits obtained from the vertical axis by the same values of t and grey arrows represent “local velocities”—vector fields of derived representations.

By contrast, the action of the third matrix from the subgroup K sharply depends on σ=ι², as illustrated by Fig. 1.2. In elliptic, parabolic and hyperbolic cases, K-orbits are circles, parabolas and (equilateral) hyperbolas, respectively. The meaning of traversal lines and vector fields is the same as on the previous figure.

Much more efficient proofs will be given later (see Exercise 2), when suitable tools will be at our disposal. It will also explain why K-orbits, which are circles, parabolas and hyperbolas, are defined by the same equation (6). Meanwhile, these formulae allow us to produce geometric characterisation of K-orbits in terms of classical notions of conic sections, cf. Appendix B.2.

The amount of similarities between orbits in the three EPH cases suggests that they should be unified one way or another. We start such attempts in the next section.

3.6 Unifying EPH Cases: The First Attempt

It is well known that any K-orbit above 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. More precisely, we define the family of double-sided right-angle cones to be parameterized by s>0:

Therefore, vertices of cones belong to the hyperbola {x=0, y²−z²=1}—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 (7). In other words, all three types of orbits are generated by the rotations of this generator along the cone.

From the above algebraic and geometric descriptions of the orbits we can make several observations.

We make a generalisation to all EPH cases of the following notion, which is well known for circles [71]*§ 2.3 and parabolas [339]*§ 10:

In the case of circles the power of a point is known as Steiner power.

3.7 Isotropy Subgroups

In Section 2.2 we described the two-sided connection between homogeneous spaces and subgroups. Section 3.3 uses it in one direction: from subgroups to homogeneous spaces. The following exercise does it in the opposite way: from the group action on a homogeneous space to the corresponding subgroup, which fixes the certain point.

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Figure 3.1: Actions of isotropy subgroups K, N′ and , which fix point ι in three EPH cases.

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Figure 3.1 shows actions of the above isotropic subgroups on the respective numbers, we call them rotations around ι. Note, that in parabolic and hyperbolic cases they fix larger sets connected with zero divisors.

It is inspiring to compare the action of subgroups K, N′ and on three number systems, this is presented on Fig. 3.2. Some features are preserved if we move from top to bottom along the same column, that is, keep the subgroup and change the metric of the space. We also note the same system of a gradual transition if we compare pictures from left to right along a particular row. Note, that Fig. 3.1 extracts diagonal images from Fig. 3.2, this puts three images from Fig. 3.1 into a context, which is not obvious from Fig. 3.2. Even greater similarity in the respective analytic expressions is presented by the next exercise.

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Figure 3.2: Actions of the subgroups K, N′, are shown in the first, middle and last columns respectively. The elliptic, parabolic and hyperbolic spaces are presented in the first, middle and last rows respectively. The diagonal drawings comprise Fig. 3.1 and the first column Fig. 1.2.

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This additional elements flips the upper and lower half-planes of double numbers—see Section 8.2. Therefore, the subgroup A″_h fixes the set {ι,−ι}.

Proof. The ax+b group acts transitively on the upper or lower half-planes. Thus, for any g∈SL₂(ℝ), there is an h in the ax+b group such that h⁻¹g either fixes ι or sends it to −ι. Thus, h⁻¹g is in the corresponding isotropy subgroup.

3.7.1 Trigonometric Functions

The actions of isotropy subgroups can be viewed as “rotations around ι”. In the Euclidean geometry rotations are transformations preserving angles. Thus, we can use isotropy subgroups to define angles in all EPH geometries as certain invariants of Möbius transformations.

As a consequence we have the following EPH form of the Pythagoras identity, cf. the identity at (a):

We continue to use the standard notation tan and tanh for the respective elliptic tan_e and hyperbolic tan_h functions, similarly for sines and cosines.

1

For the subgroup A′ this will be refined in the subsection 3.3.3.