Kinetic Brownian motion on Riemannian manifolds

Kinetic Brownian motion on Riemannian manifolds

J. Angst IRMAR, 263 Avenue du General Leclerc, 35042 RENNES, France jurgen.angst@univ-rennes1.fr http://perso.univ-rennes1.fr/jurgen.angst/ I. Bailleul IRMAR, 263 Avenue du General Leclerc, 35042 RENNES, France ismael.bailleul@univ-rennes1.fr http://perso.univ-rennes1.fr/ismael.bailleul/  and  C. Tardif LPMA, 4, Place Jussieu avenue de France, 75005 PARIS, France camille.tardif@upmc.fr http://www.proba.jussieu.fr/pageperso/tardif/
Abstract.

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle of a Riemannian manifold , collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter quantifying the size of the noise. Projection on of these processes provides random paths in . We show, both qualitively and quantitatively, that the laws of these -valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

Key words and phrases:
Diffusion processes, finite speed propagation, Riemannian manifolds, homogenization, rough paths theory, Poisson boundary
The research of the second author was partially supported by the program ANR “Retour Post-Doctorants”, under the contract ANR 11-PDOC-0025. The second author thanks the U.B.O. for their hospitality. The authors benefit from the support of Lebesgue center, ANR Labex LEBESGUE

1. Introduction

1.1. Motivations and related works

We introduce in this work a one parameter family of diffusion processes that model physical phenomena with a finite speed of propagation, collectively called kinetic Brownian motion. The need for such models in applied sciences is real, and ranges from molecular biology, to industrial laser applications, see for instance the works [CDR08, GS13, GKSW14] and the references therein. As a first step in this direction, we consider here what may be one of the simplest example of such a process and provide a detailed study of its behaviour in a fairly general geometric setting. In the Euclidean space , kinetic Brownian motion with parameter , is simply described as a random path with Brownian velocity on the unit sphere, run at speed , so

(1.1)

for some Brownian motion on the unit sphere of .

Figure 1. Kinetic Brownian motion in Euclidean space, velocity on the sphere on the left and the corresponding position on the right.

In contrast with Langevin process, whose -valued part can go arbitrarily far in an arbitrarily small amount of time, kinetic Brownian motion provides a bona fide model of random process with finite speed. Its definition on a Riemannian manifold follows the intuition provided by its version, and can be obtained by rolling on without slipping its Euclidean counterpart. Figure 2 below illustrates the dynamics of kinetic Brownian motion on the torus, as time goes on.

Figure 2. Simulations of a the kinetic Brownian motion on the torus over different time intervals.

We devote most of our efforts in this work in relating the large noise and large time behaviour of the process to the geometry of the manifold. On the one hand, we show that kinetic Brownian motion interpolates between geodesic and Brownian motions, as ranges from to , leading to a kind of homogenization. Our use of rough paths theory for proving that fact may be of independent interest. We first prove the interpolation property in the model space , and strengthen the associated convergence results into some rough paths convergence results. The twist here is that once the latter result is proved, the fact that kinetic Brownian motion can be constructed from the rough path lift of kinetic Brownian motion in by solving a rough differential equation, together with the continuity of the Itô map, in rough path topology, provides a clean justification of the homogenization phenomenon.

This result strongly echoes Bismut’s corresponding result for his hypoelliptic Laplacian [Bis05] which, roughly speaking, corresponds in its simplest features to replacing the Brownian velocity on the sphere by a velocity process given by an Ornstein-Uhlenbeck process. As a matter of fact, our method for proving the above mentionned homogenization result can also be used to recover the corresponding result for Bismut’s hypoelliptic diffusion.

On the other hand, we are able to give a complete description of the Poisson boundary of kinetic Brownian motion when the underlying Riemannian manifold is sufficiently symmetric and is fixed. This is far from obvious as kinetic Brownian motion is a hypoelliptic diffusion which is non-subelliptic. We take advantage in this task of the powerful dévissage method that was introduced recently in [AT14] as a tool for the analysis of the Poisson boundaries of Markov processes on manifolds. Its typical range of application involves a diffusion that admits a subdiffusion whose Poisson boundary is known. If the remaining piece of converges to some random variable , the dévissage method provides conditions that garantee that the invariant sigma field of will be generated by together with the invariant sigma field of , see the end of Section 3.2.2 where the main results of [AT14] are recalled. In the present situation, and somewhat like Brownian motion on model spaces, the Poisson boundary of kinetic Brownian motion is described by the asymptotic direction in which the process goes to infinity. It is remarkable, however, that depending on the geometry, kinetic Brownian motion may have a trivial Poisson boundary while Brownian motion will have a non-trivial Poisson boundary.

Kinetic Brownian motion is the Riemannian analogue of a class of diffusion processes on Lorentzian manifolds that was introduced by Franchi and Le Jan in [FLJ07], as a generalization to a curved setting of a process introduced by Dudley [Dud66] in Minkowski spacetime. These processes model the motion in spacetime of a massive object subject to Brownian fluctuations of its velocity. Despite their formal similarities, the causal structure of spacetime makes relativistic diffusions very different from kinetic Brownian motion, as the reader will find out by reading the litterature on the subject, such as [FLJ07, FLJ10, Fra09, Bai08, BR10, Bai10, Ang14], for instance.

We have organized the article as follows. Kinetic Brownian motion on a given Riemannian manifold is introduced formally in Section 1.2 below. Like its elementary -valued version, it lives in the unit tangent bundle of , and it has almost surely an infinite lifetime if the manifold is geodesically complete. Section 2 is dedicated to proving that the manifold-valued part of a time rescaled version of kinetic Brownian motion converges weakly to Brownian motion as the intensity of the noise, quantified by , increases indefinitely, if the manifold is stochastically complete. This is the main content of Theorem 2.1.2, which is proved using rough paths theory. The necessary material on this subject is recalled, so the reader can follow the proof without preliminary knowledge about rough paths. Section 3 provides a thourough description of the asymptotic behaviour of kinetic Brownian motion on a rotationnaly invariant manifold, through the identification of its Poisson boundary in a generic setting.

We collect here a number of notations that will be used throughout that work.

  • We shall use Einstein’s well-known summation convention and will denote a -dimensional oriented complete Riemannian manifold, whose unit tangent bundle and orthonormal frame bundle will be denoted respectively by and . We shall denote by a generic point of , with and , an orthonormal frame of ; we write for the canonical projection map. Last, we shall denote by the canonical basis of , with dual basis .

  • Denote by , the canonical vertical vector fields on , associated with the Lie elements of the orthonormal group of . The Levi-Civita connection on defines a unique horizontal vector field on such that , for all . The flow of this vector field is the natural lift to of the geodesic flow. Taking local coordinates on induces canonical coordinates on by writing

    Denoting by the Christoffel symbol of the Levi-Civita connection associated with the above coordinates, the vector fields and have the following expressions in these local coordinates

1.2. Definition of kinetic Brownian motion

As said above, kinetic Brownian motion on a -dimensional oriented complete Riemannian manifold , is a diffusion with values in the unit tangent bundle of . In the model setting of , it takes values in , and is described as a random path run at unit speed, with Brownian velocity, as described in Equation (1.1). As in the classical construction of Eells-Elworthy-Malliavin of Brownian motion on , it will be convenient later to describe the dynamics of kinetic Brownian motion on a general Riemannian manifold as the projection in of a dynamics with values in the orthonormal frame bundle of , obtained by rolling without splitting kinetic Brownian motion in . The following direct dynamical definition in terms of stochastic differential equation provides another description of the dynamics of kinetic Brownian motion which we adopt as a definition. The equivalence of the two point of views is shown in section 2.4.1. In the sequel will always stand for some non-negative constant which will quantify the strength of the noise in the dynamics of kinetic Brownian motion.

Definition 1.2.1.

Given , the kinetic Brownian motion with parameter , started from , is the solution to the -valued stochastic differential equation in Stratonovich form

(1.2)

started from . It is defined a priori up to its explosion time and has generator

It is elementary to see that its canonical projection on is a diffusion on its own, also called kinetic Brownian motion. In the coordinate system on induced by a local coordinate system on , kinetic Brownian motion satisfies the following stochastic differential equation in Itô form

(1.3)

where , and where is an -valued local martingale with bracket

for any and stand for the inverse of the matrix of the metric in the coordinates used here.

The readers acquainted with the litterature on relativistic diffusions will recognize in equation (1.2) the direct Riemannian analogue of the stochastic differential equation defining relativistic diffusions in a Lorentzian setting. Despite this formal similarity, the two families of processes have very different behaviours. As a trivial hint that the two situations may differ radically, note that the unit (upper half) sphere in the model space of Minkowski spacetime, is unbounded. As a result, there exists deterministic -valued paths that explode in a finite time, giving birth to exploding -valued paths . The work [Bai11] even gives some reasonnable geometric conditions ensuring the non-stochastic completeness of relativistic diffusions. No such phenomenon can happen in or on a complete Riemannian manifold for a path defined on a finite open interval, and run at unit speed.

Proposition 1.2.2 (Non-explosion).

Assume the Riemannian manifold is complete. Then kinetic Brownian motion has almost surely an infinite lifetime.

Proof.

Denote by the lifetime of kinetic Brownian motion , and assume, by contradiction, that is finite with positive probability. Since the path has unit speed, it would converge as tends to , on the event , as a consequence of the completeness assumption on . The horizontal lift of in would converge as well. Write for the Lie element of the orthonormal group of . We have then where the process satisfies the stochastic differential equation

in particular is well defined for all times . Consequently, both processes and should converge as tends to , contradicting the necessary explosion of . ∎

From now on we shall assume that the Riemannian manifold is complete, and turn in the next section to the study of kinetic Brownian motion as a function of .

2. From geodesics to Brownian paths

Let emphasize the dependence of kinetic Brownian motion on the parameter by denoting it . We show in this section that the family of laws of provides a kind of interpolation between geodesic and Brownian motions, as ranges from zero to infinity, as expressed in Theorem 2.1.2 below and illustrated in Fig. 3 below when the underlying manifold is the 2-dimensional flat torus.

Figure 3. Kinetic Brownian motion on the flat torus for .

2.1. Statement of the results

To fix the setting, add a cemetary point to , and endow the union with its usual one-point compactification topology. That being done, denote by the set of continuous paths , that start at some reference point and that stay at point if they exit the manifold . Let where stands for the filtration generated by the canonical coordinate process. Denote by the geodesic open ball with center and radius , for any . The first exit time from is denoted by , and used to define a measurable map

which associates to any path the path which coincides with on the time interval , and which is constant, equal to , on the time interval . The following definition then provides a convenient setting for dealing with sequences of random process whose limit may explode.

Definition 2.1.1.

A sequence of probability measures on is said to converge locally weakly to some limit probability on if the sequence of probability measures on converges weakly to , for every .

Equipped with this definition, we can give a precise sense to the above interpolation between geodesic and Brownian motions provided by kinetic Brownian motion.

Theorem 2.1.2 (Interpolation).

Assume the Riemannian manifold is complete. Given we have the two following asymptotics behaviours.

  • The law of the rescaled process converges locally weakly under to Brownian motion on , run at speed over the time interval , as goes to infinity.

  • The law of the non-rescaled process converges locally weakly under , as goes to zero, to a Dirac mass on the geodesic started from in the direction of the first vector of the basis .

Implicit in the above statement concerning the small noise asymptotics is the fact that the geodesic curve is only run over the time interval ; no stochastic completeness assumption is made above. The proof of the second item in the interpolation theorem is trivial, since the generator of the process then converges to the generator of the geodesic flow as it is clear from the Definition 1.2.1 of the process. We shall first prove the result stated in the first item by elementary means in the model Euclidean case, in Section 2.2. Using the tools of rough paths analysis, we shall see in Section 2.3 that elementary moment estimates allow a strengthening of the weak convergence result of Section 2.2 into the weak convergence of the rough path lift of the Euclidean kinetic Brownian motion to the (Stratonovich) Brownian rough path. To link kinetic Brownian motion in Euclidean space to its Riemannian analogue, we use the fact that the latter can be constructed by rolling on without slipping the former, using Cartan’s development map. This means, from a practical point of view, that one can construct the -valued part of kinetic Brownian motion as the solution of an -valued controlled differential equation equation in which the Euclidean kinetic Brownian motion plays the role of the control. This key fact will enable us to use the continuity of the Itô-Lyons map associated with Cartan’s development map, and transfer in Section 2.4 the weak convergence result proved for the Euclidean kinetic Brownian motion to the curved setting of any complete Riemannian manifold.

Note that the above theorem only involves local weak convergence; it can be strengthened under a very mild and essentially optimal natural assumption.

Corollary 2.1.3.

If the manifold is complete and stochastically complete then the rescaled process converges in law under , as goes to infinity, to Brownian motion run at speed over the time interval .

X.-M. Li proved recently in [Li15] an interpolation theorem similar to Corollary 2.1.3, under stronger geometric assumptions on the base manifold , requiring a positive injetivity radius and a control on the norm of the Hessian of the distance function on some geodesic ball. Her proof rests on a formulation of the weak convergence result in terms of a martingale problem, builds on ideas from homogenization theory, and uses tightness techniques. It is likely that the very robust nature of our proof, based on the rough paths machinery, offers a convenient setting for proving more general homogenization results at a low cost. As a basic illustration, notice for example that the proof below works verbatim with the Levi-Civita connection replaced by any other affine metric preserving connection . The limit process in is not in that case the lift to of a Brownian motion on anymore, but it is still described as the solution to the Stratonovich differential equation

where is an -valued Brownian motion.

Note that a slightly more general family of diffusions on the frame bundle than those given by equation (1.2) is considered in [Li15], where in addition to the Brownian noise in the canonical vertical directions, scaled by a factor , she also considers a vertical constant drift independent of . It is elementary to adapt our method to this setting.

Remark 2.1.4.

The idea of using rough paths theory for proving elementary homogenization results as in theorem 2.1.2 was first tested in the work [FGL14] of Friz, Gassiat and Lyons, in their study of the so-called physical Brownian motion in a magnetic field. That random process is described as a path in modeling the motion of an object of mass , with momentum , subject to a damping force and a magnetic field. Its momentum satisfies a stochastic differential equation of Ornstein-Uhlenbeck form

for some matrix whose eigenvalues all have positive real parts, and is a -dimensional Brownian motion. While the process is easily seen to converge to a Brownian motion, its rough path lift is shown to converge in a rough paths sense in , for any , to a random rough path different from the Brownian rough path.

The proof of the interpolation theorem 2.1.2 and Corollary 2.1.3 is split into three steps, performed in the next three subsections. We prove the Euclidean version of Corollary 2.1.3 in Section 2.2, and strengthen that weak convergence result in into a weak convergence result in rough paths topology of the rough paths lift of Euclidean kinetic Brownian motion. This is done in Section 2.3 by using some general compactness criterion on distributions in rough paths space. The point here is that kinetic Brownian motion on any complete Riemannian manifold can be constructed from its Euclidean analogue by solving a controlled differential equation in which the control is the Euclidean kinetic Brownian motion. One can then use Lyons’ universal limit theorem, on the continuity of the Itô-Lyons map, to transfer the weak convergence result of the rough kinetic Brownian motion in to its Riemannian analogue; this is explained in Section 2.4, where Corollary 2.1.3 is also proved.

2.2. Proof of the interpolation result in the Euclidean setting

When the underlying manifold is the Euclidean space , the state space of kinetic Brownian motion becomes , with coordinates , inherited from the canonical coordinates on . System (1.3) describing the dynamics of kinetic Brownian motion in a general local coordinate system takes in the present setting the simple form

(2.1)

for , with a standard -valued Brownian motion.

Proposition 2.2.1.

Given , we have the following two asymptotic regimes, in terms of .

  1. The non-rescaled process converges weakly to the Dirac mass on the geodesic path , as tends to zero.

  2. The time-rescaled process converges weakly to a Euclidean Brownian motion with covariance matrix times the identity, as tends to infinity.

Proof.

The convergence result when tends to zero is straightforward. We present two proofs of the weak convergence to Brownian motion as tends to infinity, to highlight the elementary nature of this claim. In order to simplify the expressions, let us define for all

The first approach takes as a starting point the integrated version of equation (2.1), namely

(2.2)

where is a dimensional martingale whose bracket is given by

The mid-terms in the right hand side of Equation (2.2) clearly go to zero when tends to , uniformly in , so that the asymptotic behavior of is the same as that of the martingale . To analyse that martingale, note that the time-rescaled process is a standard Brownian motion on , solution of the equation, for each

(2.3)

for some -valued Brownian motion . The bracket of the martingale is simply given in terms of by the formula

so, for a fixed time , the ergodic theorem satisfied by the process entails the almost sure convergence

The result of Proposition 2.2.1 then follows from the asymptotic version of Knight Theorem, as stated form instance in Theorem 2.3 and Corollary 2.4, pp. 524-525, in the book [RY99] of Revuz and Yor. The second approach consists in starting from the integral representation

(2.4)

where is the above standard spherical Brownian motion, so that the result can alternatively be seen a consequence of a standard central limit theorem for ergodic diffusions applied to the ergodic process , see for instance the reference [CCG12]. ∎

We shall prove in Section 2.3 that the weak convergence result of Proposition 2.2.1 can be enhanced to the weak convergence of the rough path associated with the random path to the Stratonovich Brownian rough path. This requires the study of the -valued process defined for any by the integral

For those readers not acquainted with tensor products, one can simply see the tensor product of two vectors in , as the linear map .

Proposition 2.2.2.

The -valued process converges weakly, as goes to infinity, to the Brownian rough path over , run at speed .

Proof.

It will be convenient to use the representation of given by identity (2.4), for which there is no loss of generality in assuming . Recall stands for the martingale part of in Equation (2.3), seen as an -valued path. We have

As the symmetric part of converges to the corresponding symmetric part of the Brownian rough path, by Proposition 2.2.1, we are left with proving the corresponding convergence result for the anti-symmetric part of the process , namely

(2.5)

As the process lives on the unit sphere, the terms in the first line of expression (2.5) clearly converge to zero in , as goes to infinity, uniformly on bounded intervals of time. Using the notations introduced in Equation (2.2), we are thus left with the martingale term

The weak convergence of this term to the awaited Lévy area of the Brownian rough path is dealt with by the following lemma, which concludes the proof. ∎

Lemma 2.2.3.

The -valued martingale converges weakly as goes to infinity, to the process

where is an -valued Brownian motion run at speed .

Proof.

Given , let stand for the linear map from to defined by the formula

where the star notation is used to denote linear form canonically associated with an element of the Euclidean space . With this notation, we can write

Recall that we have already proved that the process converges weakly when goes to infinity, to a Brownian motion of variance . It follows from the continuity of the map , that the process converges weakly to . Refering to Theorem 6.2, p. 383 of [JS03], the result of the lemma will follow from the previous fact if we can prove that the martingales are uniformly tight – see Definition 6.1, p. 277 of [JS03]. But since we are working with continuous martingales whose brackets converge almost surely when goes to infinity, we can use Proposition 6.13 of [JS03], p. 379, to obtain the awaited tightness. ∎

2.3. From paths to rough paths in the Euclidean setting

Proposition 2.2.2 shows that the natural lift of as a weak geometric Hölder -rough path, for , converges weakly to the Brownian rough path, when seen as an element of a space of continuous paths on , endowed with the topology of uniform convergence. We show in this section that a stronger convergence result holds, with the rough path metric in place of the uniform topology. This stronger convergence result will be the main ingredient used in Section 2.4 to prove the interpolation theorem 2.1.2, by relying on Lyons’ universal limit theorem.

For the basics of rough paths theory, we refer the reader to the very nice account given in the book [FH14] by Friz and Hairer. Alternative pedagogical accounts of the theory can be found in [Bai14a, Bau13, Lej09], see also the book [FV10] of Friz and Victoir for a thourough account of their approach of the theory. Let be given.

Proposition 2.3.1.

The weak geometric Hölder -rough path converges weakly as a rough path to the Brownian rough path.

The remainder of this section is dedicated to the proof of this statement. Our strategy of proof is simple. Using elementary moment estimates, we show that the family of laws of is tight in some rough paths space. As the rough path topology is stronger than the topology of uniform convergence on bounded intervals, Proposition 2.2.2 identifies the unique possible limit for these probability measures on the rough paths space, giving the convergence result as a consequence.

To show tightness, we shall rely on the following Kolmogorov-Lamperti-type compactness criterion; see Corollary A.12 of [FV10] for a reference. As a shortcut, we will write or for the increment , for any , and any path with values in a vector space.

Theorem 2.3.2 (Kolmogorov-Lamperti tightness criterion).

Given any , consider the laws of , for , as probability measures on the metric space of weak geometric -Hölder rough paths. If the following moment estimates

(2.6)

hold for all , for some positive constants , for all , then the family of laws of is tight in .

We shall prove these two bounds separately, starting with the bound on . We use for that purpose the representation

in terms of a unit speed Brownian motion on the sphere, already used in the previous section, together with Equation (2.3) giving the dynamics of in terms of an -valued Brownian motion . Denoting by the coordinate of a vector , this gives the equation

So we have

(2.7)

for some positive constant depending only on . Note that term contains an implicit sum over and . We use Burkholder-Davis-Gundy inequality to deal with it, taking into account the fact that the process takes values in the unit sphere. This gives the estimate

(2.8)

For term , remark simply that since the identity

holds for all , for some -valued Brownian motion , we have

using the fact that lives on the unit sphere, so the process is a real-valued Brownian motion, together with Burkholder-Davis-Gundy inequality. So we have

for all , for some well-chosen constant , since . The upper bound , follows then from equations (2.7) and (2.8). To deal with the double integral , let us start from Equation (2.3) to write

(2.9)

Again, as the process lives in the unit sphere, the first term in the right hand side above is bounded above by . Let us denote by the path in with values in whose velocity is given . To deal with the term involving the martingale in equation (2.9), we use an integration by parts, and the elementary inequality , to get the existence of a positive constant depending only on such that we have

(2.10)

where we have set

There is an absolute positive constant for which

On the one hand, taking once more into account the fact the the process lives on the unit sphere, the Burkholder-Davis-Gundy inequality gives us the upper bound

On the other hand, we can use the identity

and the fact that has the same law as , to see that

We recognize in the first term on the right hand side term , with in the role of , and use the Burkholder-Davis-Gundy inequality to deal with the other expectation. The two results together give an upper bound of size , up to a multiplicative constant depending only on , showing that is also bounded above by a constant multiple of . The inequality of Burkholder-Davis-Gundy is used once more, together with the bound just proved, to estimate term by a constant multiple of

Note that all the above estimates hold regardless of the values of . All together, these estimates prove that the two conditions of the compactness in Theorem 2.3.2 hold for the family of weak geometric -Hölder rough paths . The family of laws of is thus relatively compact on the space . As its unique possible cluster point is identified by Proposition 2.2.2, this proves the weak convergence of the random weak geometric -Hölder rough paths to the Brownian rough path, as stated in Proposition 2.3.1.

2.4. From the Euclidean to the Riemannian setting via Cartan’s development map

The homogenization result proved in Proposition 2.3.1 puts us in a position to use the machinery of rough differential equations and prove homogenization results for solutions of rough differential equations driven by , using to our advantage the continuity of the Itô map in a rough paths setting. This is in particular the case of kinetic Brownian motion on any complete Riemannian manifold, which can be constructed from kinetic Brownian motion on , using Cartan’s development map. The interpolation theorem 2.1.2 will follow from this picture of kinetic Brownian motion as a continuous image of . Before following that plan, we recall the reader the basics of Cartan’s development method and rough differential equations.

2.4.1. Cartan’s development map

As advertized above, one can actually construct kinetic Brownian motion on a complete Riemannian manifold by rolling on without slipping its Euclidean analogue. While this will be clear to the specialists from the very definition of kinetic Brownian motion, Cartan’s development procedure can be explained to the others as follows. In its classical form, this machinery provides a flexible and convenient way of describing paths on from -valued paths; it requires the use of the frame bundle and the horizontal vector fields associated with a choice of connection (Levi-Civita connection presently).

Let H stand for the -valued -form on uniquely characterized by the property

for any and .

Definition 2.4.1.

Given in , Cartan’s development of an -valued path of class is defined as the solution to the ordinary differential equation

(2.11)

started from . As before, we shall write .

This description of an -valued path may seem somewhat different from the kind of dynamics described by Equation (1.2) defining kinetic Brownian motion. To make the link clear, assume, without loss of generality, that is run at unit speed, and denote its speed by . Recall the definition of the elements of the Lie algebra of given at the end of Section 1.1. Then, given an orthonormal basis of , with