Lecture 11 Unitary Rotations

One of the important advantages of the elliptic and hyperbolic unit disks introduced in Sections 10.1–10.2 is a simplification of isotropy subgroup actions. Indeed, images of the subgroups K and , which fix the origin in the elliptic and hyperbolic disks, respectively, consist of diagonal matrices—see (4) and (10). These diagonal matrices produce Möbius transformations, which are multiplications by hypercomplex unimodular numbers and, thus, are linear. In this chapter, we discuss the possibility of similar results in the parabolic unit disks from Section 10.3.

11.1 Unitary Rotations—an Algebraic Approach

Consider the elliptic unit disk zz<1 (6) with the Möbius transformations transferred by the Cayley transform (1) from the upper half-plane. The isotropy subgroup of the origin is conjugated to K and consists of the diagonal matrices (

) (4). The corresponding Möbius transformations are linear and are represented geometrically by rotation of ℝ² by the angle 2φ. After making the identification ℝ²=ℂ, this action is given by the multiplication e^{2i φ}. The rotation preserves the (elliptic) distance (5) given by

Therefore, the orbits of rotations are circles and any line passing the origin (a “spoke”) is rotated by an angle 2φ—see Fig. 11.1(E). We can also see that those rotations are isometries for the conformally-invariant metric (14) on the elliptic unit disk. Moreover, the rotated “spokes”—the straight lines through the origin—are geodesics for this invariant metric.

A natural attempt is to employ the algebraic aspect of this construction and translate to two other cases (parabolic and hyperbolic) through the respective hypercomplex numbers.

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Figure 11.1: Rotations of algebraic wheels, i.e. multiplication by eι t: elliptic (E), trivial parabolic (P0) and hyperbolic (H). All blue orbits are defined by the identity x2−ι2y2=r2. Green “spokes” (straight lines from the origin to a point on the orbit) are “rotated” from the real axis.

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The value of e^{ι t} can be defined, e.g. from the Taylor expansion of the exponent. In particular, for the parabolic case, ε^k=0 for all k≥ 2, so e^{ε t}=1+ε t. Then, the parabolic rotations explicitly act on dual numbers as follows:

In other words, the value of the imaginary part does not affect transformation of the real one, but not vice versa. This links the parabolic rotations with the Galilean group [339] of symmetries of classical mechanics, with the absolute time disconnected from space, cf. Section 8.4.

The obvious algebraic similarity from Exercise 1 and the connection to classical kinematics is a widespread justification for the following viewpoint on the parabolic case, cf. [129, 339]:

• The parabolic trigonometric functions cosp and sinp are trivial:

  cosp t =± 1,    sinp t=t. (3)

• The parabolic distance is independent from v if u≠ 0:

  u2=(u+ε v)(u−ε v). (4)

• The polar decomposition of a dual number is defined by, cf. [339]*App. C(30’):

  u+ε v = u(1+ε

  v u

  ),    thus

  ⎪ ⎪

  u+ε v

  ⎪ ⎪

  =u,   arg(u+ε v)=

  v u

  . (5)

• The parabolic wheel looks rectangular, see Fig. 11.1(P₀).

These algebraic analogies are quite explicit and are widely accepted as an ultimate source for parabolic trigonometry [241, 129, 339]. However, we will see shortly that there exist geometric motivation and a connection with the parabolic equation of mathematical physics.

11.2 Unitary Rotations—a Geometrical Viewpoint

We make another attempt at describing parabolic rotations. The algebraic attempt exploited the representation of rotation by hypercomplex multiplication. However, in the case of dual numbers this leads to a degenerate picture. If multiplication (a linear transformation) is not sophisticated enough for this, we can advance to the next level of complexity: linear-fractional.

In brief, we change our viewpoint from algebraic to geometric. Elliptic and hypercomplex rotations of the respective unit disks are also Möbius transformations from the one-parameter subgroups K and in the respective Cayley transform. Therefore, the parabolic counterpart corresponds to Möbius transformations from the subgroup N′.

For the sake of brevity, we will only treat the elliptic version P_e of the parabolic Cayley transform from Section 10.3. We use the Cayley transform defined by the matrix

The Cayley transform of matrices (2) from the subgroup N is

This is not too different from the diagonal forms in the elliptic (4) and hyperbolic (10). However, the off-diagonal (1,2)-term destroys harmony. Nevertheless, we will continue defining a unitary parabolic rotation to be the Möbius transformation with the matrix (6), which is no longer multiplication by a scalar. For the subgroup N′, the matrix is obtained by transposition of (6).

In the elliptic and hyperbolic cases, the image of reference point (−ι) is:

This coincides with the cyclic rotations defined in [339]*§ 8. A comparison with the Euler formula seemingly confirms that sinp t=t, but suggests a new expression for cosp t:

Therefore, the parabolic Pythagoras’ identity would be

which fits well between the elliptic and hyperbolic versions:

The identity (10) is also less trivial than the version cosp² t =1 from (3)–(4)—see also [129]. The possible ranges of the cosine and sine functions are given by the table:

There is a second option for defining parabolic rotations for the lower-triangular matrices from the subgroup N′. The important difference is now that the reference point cannot be −ε since it is a fixed point (as well as any point on the vertical axis). Instead, we take ε⁻¹, which is an ideal element (a point at infinity), since ε is a divisor of zero. The proper treatment is based on the projective coordinates, where point ε⁻¹ is represented by a vector (1, ε)—see Section 8.1.

A comparison with (9) shows that this form is obtained by the substitution t↦ t⁻¹. The same transformation gives new expressions for parabolic trigonometric functions. The parabolic “unit cycle” is defined by the equation u²−v=1 for both subgroups—see Fig. 10.1(P) and (P′) and Exercise 4. However, other orbits are different and we will give their description in the next section. Figure 10.1 illustrates Möbius actions of matrices (7), (8) and (6) on the respective “unit disks”, which are images of the upper half-planes under the respective Cayley transforms from Sections 10.1 and 10.3.

At this point, the reader may suspect that the structural analogy mentioned at the beginning of the section is insufficient motivation to call transformations (9) and (11) “parabolic rotation” and the rest of the chapter is a kind of post-modern deconstruction. To dispel any doubts, we present the following example:

The last example hints at further works linking the parabolic geometry with parabolic partial differential equations.

11.3 Rebuilding Algebraic Structures from Geometry

Rotations in elliptic and hyperbolic cases are given by products of complex or double numbers, respectively, and, thus, are linear. However, non-trivial parabolic rotations (9) and (11) (Fig. 10.1(P) and (P′)) are not linear.

Can we find algebraic operations for dual numbers which linearise these Möbius transformations? To this end, we will “revert a theorem into a definition” and use this systematically to recover a compatible algebraic structure.

11.3.1 Modulus and Argument

In the elliptic and hyperbolic cases, orbits of rotations are points with a constant norm (modulus), either x²+y² or x²−y². In the parabolic case, we already employed this point of view in Chapter 9 to treat orbits of the subgroup N′ as equidistant points for certain Möbius-invariant metrics, and we will do this again.

The only straight lines preserved by both parabolic rotations N and N′ are vertical lines, so we will treat them as “spokes” for parabolic “wheels”. Elliptic spokes, in mathematical terms, are “points on the complex plane with the same argument”. Therefore we again use this for the parabolic definition.

Both Definitions 6 and 8 possess natural properties with respect to parabolic rotations.

11.3.2 Rotation as Multiplication

We revert again theorems into definitions to assign multiplication. In fact, we consider parabolic rotations as multiplications by unimodular numbers, thus we define multiplication through an extension of properties from Exercise 9:

We also need a special form of parabolic conjugation which coincides with sign reversion of the argument:

Obviously, we have the properties | w |^(′)=| w |^(′) and arg^(′)w=−arg^(′) w. A combination of Definitions 6, 8 and 11 uniquely determines expressions for products.

Although both the above expressions look unusual, they have many familiar properties, which are easier to demonstrate from the implicit definition rather than the explicit formulae.

In particular, the property (3) will be crucial below for an inner product.

We defined multiplication though the modulus and argument described in the previous subsection. Our notion of the norm is rotational-invariant and unique up to composition with a monotonic function of a real argument—see the discussion in Section 9.2. However, the argument can be defined with greater freedom—see Section 11.6.2.

11.4 Invariant Linear Algebra

Now, we wish to define a linear structure on ℝ² which is invariant under point multiplication from the previous subsection (and, thus, under the parabolic rotations, cf. Exercise 2). Multiplication by a real scalar is straightforward (at least for a positive scalar)—it should preserve the argument and scale the norm of a vector. Thus, we have formulae, for a>0,

On the other hand, the addition of vectors can be done in several different ways. We present two possibilities—one is tropical and the other, exotic.

11.4.1 Tropical form

The above example is fundamental in the broad area of tropical mathematics or idempotent mathematics, also known as Maslov dequantisation algebras—see [246] for a comprehensive survey.

Let us introduce the lexicographic order on ℝ²:

One can define functions min and max of a pair of points on ℝ², respectively. Then, an addition of two vectors can be defined either as their minimum or maximum.

Although an investigation of this framework looks promising, we do not study it further for now.

11.4.2 Exotic form

Addition of vectors for both subgroups N and N′ can be defined by the following common rules, where subtle differences are hidden within the corresponding Definitions 6 (norms) and 8 (arguments).

The rule for the norm of sum (22) may look too trivial at first glance. We should say, in its defence, that it fits well between the elliptic | w+w′ |≤ | w |+| w′ | and hyperbolic | w+w′ |≥ | w |+| w′ | triangle inequalities for norms—see Section 9.3 for their discussion.

The rule (21) for the argument of the sum is also not arbitrary. From the law of sines in Euclidean geometry, we can deduce that

where ψ^(′)=argw^(′) and φ=arg (w+w^(′)). Using parabolic expression (3) for the sine, sinp θ=θ, we obtain the arguments addition formula (21).

A proper treatment of zeros in the denominator of (21) can be achieved through a representation of a dual number w=u+ε v as a pair of homogeneous polar (projective) coordinates [a,r]=[ | w |^(′) · arg^(′) w, | w |^(′)] (primed version for the subgroup N′). Then, the above addition is defined component-wise in the homogeneous coordinates

The multiplication from Definition 11 is given in the homogeneous polar coordinates by

Thus, homogeneous coordinates linearise the addition (21) and (22) and multiplication by a scalar (19).

Both formulae (21) and (22) together uniquely define explicit expressions for the addition of vectors. However, these expressions are rather cumbersome and not really much needed. Instead, we list the properties of these operations.

To complete the construction, we need to define the zero vector and the inverse. The inverse of w has the same argument as w and the opposite norm.

Thereafter, we can check that scalar multiplications by negative reals are given by the same identities (19) and (20) as for positive ones.

11.5 Linearisation of the Exotic Form

Some useful information can be obtained from the transformation between the parabolic unit disk and its linearised model. In such linearised coordinates (a,b), the addition (21) and (22) is done in the usual coor­dinate­-wise manner: (a,b)+(a′,b′)=(a+a′,b+b′).

We also note that both norms (14) have exactly the same value a+b in the respective (a,b)-coordinates. It is not difficult to transfer parabolic rotations from the (u,v)-plane to (a,b)-coordinates.

This should not be surprising, since any associative and commutative two-di­men­sio­nal algebra is formed either by complex, dual or double numbers [241]. However, it does not trivialise our construction, since the above transition is essentially singular and shall be treated within the birational geometry framework [224]. Similar singular transformations of time variable in the hyperbolic setup allow us to linearise many non-linear problems of mechanics [281, 282].

Another application of the exotic linear algebra is the construction of linear representations of SL₂(ℝ) induced by characters of subgroup N′ which are realised as parabolic rotations [194].

11.6 Conformality and Geodesics

We conclude our consideration of the parabolic rotations with a couple of links to other notions considered earlier.

11.6.1 Retrospective: Parabolic Conformality

The irrelevance of the standard linear structure for parabolic rotations manifests itself in many different ways, e.g. in an apparent “non-conformality” of lengths from parabolic foci, that is, with the parameter σ_r=0 in Proposition 3. Adapting our notions to the proper framework restores a clear picture.

The initial Definition 13 of conformality considers the usual limit y′→ y along a straight line, i.e. a “spoke” in terms of Fig. 11.1. This is justified in the elliptic and hyperbolic cases. However, in the parabolic setting the proper “spokes” are vertical lines—see Definition 8 of the argument and the illustration in Fig. 10.1(P) and (P′). Therefore, the parabolic limit should be taken along the vertical lines.

11.6.2 Perspective: Parabolic Geodesics

We illustrated unitary rotations of the unit disk by an analogy with a wheel. Unitarity of rotations is reflected in the rigid structure of transverse “rims” and “spokes”. The shape of “rims” is always predefined—they are orbits of rotations. They also define the loci of points with the constant norm. However, there is some flexibility in our choice of parabolic “spokes”, i.e. points with the same value of argument. This is not determined even under the strict guidance of the elliptic and hyperbolic cases.

In this chapter, we take the simplest possible assumption: elliptic and hyperbolic “spokes” are straight lines passing through the origin. Consequently, we have looked for parabolic “spokes” which are straight lines as well. The only family of straight lines invariant under the parabolic rotations (9) and (11) are vertical lines, thus we obtained the situation depicted in Fig. 10.1.

However, the above path is not the only possibility. We can view straight lines in the elliptic and hyperbolic cases as geodesics for the invariant distance on the respective unit disks, see Exercise 14(e,p). This is perfectly consistent with unitary rotations and suggests that parabolic “rims” for rotations (11) shall be respective parabolic geodesics on the parabolic unit disk, see Fig. 11.2.

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Figure 11.2: Geodesics as spokes. In each case “rims” are points equidistant from the origin, “spokes” are geodesics between the origin and any point on the rims.

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There are three flavours of parabolic geodesics, cf. Fig. 9.3(P_E–P_H) and Exercise 14(p). All of them can be used as “spokes” of parabolic unit disk, but we show only the elliptic flavour in Fig. 11.2.