Weak Lie symmetry and extended Lie algebraHubert Goenner Citation: J. Math. Phys. 54, 041701 (2013); doi: 10.1063/1.4795839 View online: http://dx.doi.org/10.1063/1.4795839 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i4 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS 54, 041701 (2013)

Weak Lie symmetry and extended Lie algebraa)

Hubert Goennerb)

Institute for Theoretical Physics, Friedrich-Hund-Platz 1, University of Goettingen, D-37077Gottingen, Germany

(Received 11 December 2012; accepted 4 March 2013; published online 1 April 2013)

The concept of weak Lie motion (weak Lie symmetry) is introduced. Applicationsgiven exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group.In this context, a particular generalization of Lie algebras is found (“extended Liealgebras”) which turns out to be an involutive distribution or a simple example for atangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such analgebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroidsand to Lorentz geometries constructed on them (1-dimensional gravitational fields).C© 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795839]

I. INTRODUCTION

In the spirit of Felix Klein’s Erlangen program (cf. Refs. 1 and 2), several new concepts willbe introduced: weak (Lie) motions (cf. Sec. IV) and groups of extended motions (cf. Sec. VIII).The last concept is related to a suggested widening of the physicists’ concept of a Lie algebra toparticular tangent Lie algebroids, called here extended Lie algebras. Some of the corresponding finitetransformations are presented: they are no longer Lie groups. Also, an extension of the Cartan-Killingform will be proposed which up to now seemingly has not been studied. Its definition allows theintroduction of Riemannian and Lorentz metrics on the sections of a subbundle of the tangent bundle.The mathematical literature for algebroids and groupoids3 has lead to some formal applications toLagrangian mechanics.4–6 The particular tangent Lie algebroids presented here are an example forsuch structures much closer to physics than the examples usually given by mathematicians. Theyalready emerged in pre-relativistic relative mechanics.

II. LIE-DRAGGING

A. Lie algebra for physicists

In metric geometry, the concept of symmetry is expressed by an isometry of the metrical tensorgab of such a space. This means that this tensor field remains unchanged along the flow of a vectorfield X. This demand may be formulated by help of the Lie derivative defined for tangent vectorfields X := ξ a ∂

∂xa , Y := ηa ∂∂xa by

LX Y = [X, Y ], (1)

where [., .] denotes the Lie-bracket [A, B] = AB − BA. If (1) is expressed by the components ξ a, ηa

of the tangent vectors X, Y, then7

Lξ ηa = ηa

,cξc − ηcξ a

,c, (2)

a)A summary of this article has been presented at the “90th Encounter between Mathematicians and Theoretical Physicists”at the Institut de Recherche Mathematique Avancee (University of Strasbourg and CNRS), September 20–22, 2012.

b)E-mail: goenner@theorie.physik.uni-goettingen.de

0022-2488/2013/54(4)/041701/15/$30.00 C©2013 American Institute of Physics54, 041701-1

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041701-2 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

where ηa,c = ∂ηa

∂xc . If LX Y = 0, the vector field X is called a symmetry of the vector field Y. Suchsymmetries play an important role for the integration of differential equations.8 The Leibniz ruleholds for the Lie derivative. From (1) we have

LZLX Y = [Z , [X, Y ]], (3)

and with help of the Jacobi identity:

LZLX Y + LYLZ X + LXLY Z = [Z , [X, Y ]] + [Y, [Z , X ]] + [X, [Y, Z ]] = 0. (4)

From (4):

LZLX Y − LXLZ Y = [[X, Z ], Y ] = L[X,Z ]Y = LLX Z Y. (5)

For a Lie group, a special subspace of the tangent space is formed by the infinitesimal generatorsX (i) := ξ a

(i)∂

∂xa , (i, j, l = 1, 2, . ., p) by means of the Lie-bracket

[X (i), X ( j )] = cli j X (l), (6)

with structure constants cli j . It is named a Lie algebra.9 From (6) follows:

LXiLX j Xk = cljkcm

il Xm (7)

such that according to (4):

cljkcm

il + cli j c

mkl + cl

ki cmjl = 0. (8)

A symmetric bilinear form, the Cartan-Killing form, may be introduced,

σi j := cmil cl

jm . (9)

If it is nondegenerate, i.e., for semisimple Lie groups, σ ij can be used as a metric in group space. InSec. VII, we will permit that the structure constants become directly dependent on the componentsξ a

i of the vector fields Xi(x): they will become structure functions.

B. Lie-dragging

Under “Lie-dragging” with regard to an arbitrary C∞ vector field X = ξ a ∂∂xa we understand the

operation of the Lie derivative on any geometric object without the simultaneous requirement thatthe result be zero. Applied to the metric gab, this means

Lξ gab = γab, (10)

where γ ab is a symmetric tensor of any rank between 0 and n (in n-dimensional space). In the sequelwe will be interested in the case γ ab �= λgab. This use of the name “Lie-dragging” is different fromthe one in Ref. 10. By (1), the Lie-dragging of a vector field is expressed.

For a tensor field, Lie-dragging neither conserves the rank of the field, nor, if it is applied to asymmetric bilinear form, its signature. The vector fields X might be classified according to whetherLie-dragging with them leads to a prescribed rank for given metric gab. In any case, not everyarbitrary γ ab can be reached by Lie-dragging (cf. the Appendix).

Equation (10) can be read in different ways:(A) Given a single vector field (a set of vector fields) and an arbitrary metric gab; the set of

all possible bilinear forms γ ab is to be determined by a straightforward calculation. This is anintermediate step for the determination of weak Lie motions of gab.

(B) Given a single vector field (a set of vector fields) and a fixed target tensor γ ab; the metricsgab which are Lie-dragged into it are to be determined. This requires solving a system of 1st-orderPDEs.

(C) Given both a start metric gab and a target metric γ ab. The task is to determine the vectorfields X dragging the one into the other.

If we ask for both, LX gab = γab and LXγab = gab, then we are back to weak homotheticmappings for both g and γ . Cf. Sec. III.

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041701-3 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

For a first example for Lie-dragging in space-time leading to tensors of lower rank, we look atthe Kasner metric:

ds2 = (dx0)2 − (x0)2p1 (dx1)2 − (x0)2p2 (dx2)2 − (x0)2p3 (dx3)2, (11)

an exact solution of Einstein’s vacuum field equations if p1 + p2 + p3 = 1 = (p1)2 + (p2)2

+ (p3)2, p1, p2, p3 constants. Lie-dragging with

X = δa0

∂

∂xa

after a coordinate change leads to the space sections,

ds2 = −(y0)2p1 (dy1)2 − (y0)2p2 (dy2)2 − (y0)2p3 (dy3)2.

Unlike this, Lie-dragging of (11) with

X = f (x0)δa1

∂

∂xa

leads to a tensor of rank 2 :11 γab = 2 d f (x0)dx0 g1(aδ

0b).

In the second example, a Lie-dragged metric of rank 1 is prescribed. Let

Lξ gab = Xa Xb, (12)

with the vector field X tangent to a null geodesic:

(g∇b Xa)Xb = 0, gab Xa Xb = 0. (13)

From the definition of Lξ gab given in (16) and (13), (Xsξ s), aXa = 0 follows: Xsξ s must beconstant along the geodesic. Equation (13) leads to a restriction on ξ for given null geodesic, or forXa if the vector field ξ is given. Xa generates a powerful-weak motion (cf. Sec. IV).

III. MOTIONS AND COLLINEATIONS

On a manifold with differentiable metric structure, a motion is defined by the vanishing of theLie-derivative of the metric with regard to the tangent vector field X = ξ a ∂

∂xa , i.e., by Lie-dragginginto zero:

LX g(Y, Z ) = 0 = Xg(Y, Z ) + g(Z ,LX Y ) + g(Y,LX Z )

= Xg(Y, Z ) + g(Z , [X, Y ]) + g(Y, [X, Z ]), (14)

where X, Y, Z are tangent vector fields. In local coordinates, (14) reads as

γab = Lξ gab = 0 = gab,cξc + gcbξ

c,a + gacξ

c,b, (15)

with gab = gba. The vector field ξ is named a Killing vector; its components generate an infinitesimalsymmetry transformation:12 xi → xi ′ = xi + ξ i . (15) may be expressed in a different form:

Lξ gab = 2g∇ (aξb) = 0. (16)

In (16),g∇ is the covariant derivative with respect to the metric gab (Levi Civita connection), and ξ a

= gabξb. From (15) we can conclude that Lξ ds = 0 for all dxa, i.e., all distances remain invariant.

A consequence of (15) is that the motions ξ form a Lie group and the corresponding infinitesimalgenerators X (i) := ξσ

(i)∂

∂xσ a Lie algebra (6), cf. Ref. 14.As an example for a group of motions in 3-dimensional Euclidean space, a Lie group G3 acting

on V3 with finite transformations is used,

x1′ = x1 + c1, x2′ = x2 + c2x1, x3′ = x3 + c3. (17)

The corresponding Lie algebra is13

[X1, X2] = 0, [X1, X3] = 0, [X2, X3] = X1. (18)

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041701-4 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

In quantum mechanics, the algebra is known under the name Heisenberg algebra and, in the form[p, q] = 1z, [p, z] = 0, [q, z], for operator-valued elements p, q and the unit operator is reflecting thecanonical commutation relations.

Lie-dragging with the vector fields ξ a1 = δa

2 , ξ a2 = δa

3 , ξ a3 = −δa

1 + x3ξ a2 gives

Lξ1 gab = gab,2 =:(1)γ ab,Lξ2 gab = gab,3 =:

(2)γ ab,

Lξ3 gab = −gab,1 + x3gab,2 + 2g2(aδ3b) =:

(3)γ ab. (19)

All(i)γ ab can have full rank. The demand

(i)γ ab = 0, i = 1, 2, 3, makes this G3 a group of motions

whence follows:

gab =

⎛⎜⎜⎝

α(0)11 α

(0)12 P1

α(0)21 α

(0)22 P ′

1

P1 P ′1 P2

⎞⎟⎟⎠,

where P1 = α(0)12 x1 + α

(0)13 , P ′

1 = α(0)22 x1 + α

(0)23 , and P2 = α

(0)22 (x1)2 + 2α

(0)23 x1 + α

(0)33 with

α(0)33 , α

(0)1p , α

(0)2p , (p = 1, 2, 3) constants. We will see in Sec. V C how the metric looks if the

group is demanded to be a complete set of weak (Lie) motions.Further types of symmetries are defined by the vanishing of the Lie derivative applied to other

geometric objects like connection (“affine collineations” Lξcab(g) = 0, cf. Ref. 15, curvature tensor

(“curvature collineations” Lξ Rcdab(g) = 0, cf. Ref. 16, Ricci tensor (“Ricci” or “contracted curvature

collineations”Lξ Rcabc(g) = 0, cf. Ref. 17. Another generalization is the concept of conformal Killing

vector, defined by

Lξ gab = λ(x1, ..xn)gab. (20)

A subcase are homothetic motions with λ = λ0 = const. Conformal Killing vectors are included inwhat follows.

IV. WEAK LIE MOTIONS (WEAK SYMMETRIES)

In the 1980s, a concept of “p-invariance” has been introduced,18

Lξ . . . . . .Lξ gab = 0, (21)

with p Lie derivatives, p > 1, acting on the metric. At the time, for p = 2 an application has beengiven in Einstein-Maxwell theory.19 In the following we will concentrate on this case p = 2.

Definition 1: An infinitesimal point transformation x → x + ξ satisfying

LξLξ gab = 0,Lξ gab �= 0, (22)

generates a “weak Lie motion.”A coordinate-free formulation of (22) is

LWLZ g(X, Y ) = [W, Z ]g(X, Y ) − g([W, [Z , X ]], Y ) − g(X, [Y, [W, Z ]]).

If applied to other geometric objects, we call (22) “weak symmetry.” In the set of solutionsof (22), the isometries (motions) must also occur. We speak of genuine weak Lie motions whenmotions are to be excluded. The expression weak isometry is also used. Equation (22) can be readin two ways:

- The metric gab is given; determine the generator ξ of a weak Lie motion;- A vector field or a Lie algebra is given; determine the metric gab which allows these fields as

weak Lie motions.As has been pointed out in Ref. 18, a disadvantage of the new concept is that LξLξ gab = 0 does

not follow from LξLξ gab = 0 for Lξ gab �= 0. In fact,

LξLξ gab = −gas gbtLξLξ gst + 2gat gbpgsq (Lξ gpq )(Lξ gst ). (23)

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041701-5 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

Consequently, in general LξLξ gab = 0 and LξLξ gab = 0 define slightly different invariance con-cepts. If both conditions are imposed, Lξ gab = �(x)kakb with the null vector ka(grskrks = 0), andarbitrary scalar function � follows. In this case, we call the weak motion generated by X = ξ a ∂

∂xa amore “powerful” weak motion. It entails the existence of a null vector ka with Lξ ka = −kaLξ (ln�).What is called here “powerful” motion, would have be named cosymmetric-2-invariance inRef. 18, p. 138. In Euclidean space Lξ gab = 0 results. For p > 2 the situation would becomestill more complicated.

A. Weak symmetries

That a weak symmetry can be really weaker than a symmetry is seen already when the Liederivative is applied twice to a function f(x1, . . . xn):

LXLX f = LξLξ f = X X f!= 0. (24)

In n-dimensional Euclidean space Rn, for a translation in the direction of the k-axis with ξ i = δi(k),

we obtain from (24) f = xkf1(x1, .., xk − 1, xk + 1, .., ..xn) + f2(x1, .., xk − 1, xk + 1, .., ..xn) in place

of f = f(x1, .., xk − 1, xk + 1, .., ..xn) for Lξ f!= 0. For the full translation group of Rn, (24) leads

to a polynomial of degree n in the variables (x1, .., xn) with constant coefficients and linear ineach variable (x1, .., xn). Thus, for n = 3, f = c123x1x2x3 + �3

r,s=1;r<s,crs xr xs + �3s=1cs xs + c0

as compared to f = f0 for the translation group as a group of motions. This result follows only ifDefinition 3 for a complete set of weak symmetries is applied, cf. Sec. IV B.

For a rotation Rik = xi ∂

∂xk − xk ∂∂xi (i, k fixed), a function satisfying LξLξ f

!= 0 is given by

f = α1(x1, .., xi−1, xi+1, ..xk−1, xk+1, ..xn) × arctan xi

xk + α2(x1, .., xi − 1, xi + 1, ..xk − 1, xk + 1, ..xn),with Lξ f = −α1 �= 0 for this rotation. For the full rotation group SO(3) in 3-dimensional space,f = f (r ), r =

√(x1)2 + (x2)2 + (x3)2 follows: no genuine weak motion is possible in this case.

These examples show that the set of weak-Lie invariant functions can be larger.An enlargement of a subgroup of the Abelian translation group in an n-dimensional euclidean

space is given by

x1′ = x1 + G1(xk+1, .., xn), .., xk ′ = xk + Gk(xk+1, .., xn), x (k+1)′ = xk+1, .., xn′ = xn, (25)

with arbitrary C∞ functions G1, G2, .., Gk. Weak Lie symmetry under this group for the functionf(x1, .., xn) leads to the same result as for the translation group, although (25) no longer is a Liegroup.

A link between weak Lie symmetry of scalars and weak Lie motions can be found in conformallyflat metrics: gab = f(x1, x2, .., xn)ηab due to

LξLξ gab = (LξLξ f )ηab + 2Lξ f Lξ ηab + LξLξ ηab. (26)

In the special case of (20) follows

LξLξ gab = (λ2 + λ,sξs)gab,LξLξ gab = (λ2 − λ,sξ

s)gab. (27)

Hence, in this case nothing new is obtained by letting the Lie-derivative act twice. The concept ofconformal Killing vector could also be weakened to weak conformal Killing vector by the demand,

LξLξ gab = λ(xi )gab,Lξ gab �= μ(x j )gab. (28)

B. Complete sets of weak Lie motions

If gab allows the maximal group of motions with (n+12 ) parameters, no genuine weak Lie motions

do exist. If gab allows a r-parameter group of motions, then (n+12 ) − r genuine weak Lie motions may

exist. The case of a Lie group with (n+12 ) − 1 parameter acting as an isometry group cannot occur in

n-dimensional space.20 Hence, in space-time which allows a 10-parameter group as maximal group,

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041701-6 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

no 9-parameter Lie group exists. For 4-dimensional Lorentz-space (with signature ± 2), 8-parameterLie groups are likewise excluded as isometry groups.21 This does not hold for Finsler geometry bywhich an 8-parameter Lie group is admitted. Cf. Refs. 22–24. Thus, besides the maximal group, thelargest group of motions in space-time is a 7-parameter group. Petrov’s claim that for 4-dimensionalLorentz spaces 7-parameter Lie groups are excluded, is not correct, cf. Ref. 13, p. 134 and Ref. 25,p. 122). In this case, the largest group of weak Lie motions would then be a 3-parameter Lie group.

According to (5), a consequence for weak motions is

(LξiLξ j − Lξ jLξi )gab = L(Lξi ξ j )gab = Lckji ξk

gab = ckjiLξk gab. (29)

Equation (29) provides a hint about how a group of weak Lie symmetries is to be defined whena set of vector fields, ξ , η, ζ , .. has been found satisfying (22). For genuine weak motions, notall of the following equations can be satisfied: LηLξ gab = 0,LξLηgab = 0,LηLζ gab = 0,LζLηgab

= 0,LζLξ gab = 0,LξLζ gab = 0, . . . . If the r vectors ξ (k), k = 1, 2, .., r are the infinitesimalgenerators of a Lie, group, the above demand in general leads into an impasse: instead of itsintended role as a weak Lie-invariance group, it reduces to an isometry group. This is due to (5) or(29). An exception holds if some of the vector fields commute.

Consequently, the following definition may be introduced:

Definition 2 (Strong complete set):A Lie algebra presents a strong complete set of weak Lie symmetries if at least one of the

corresponding Lie algebra elements does not generate a motion (Lξ( j) gab �= 0 for one (j), at least)and the following (m+1

2 ), m > 1 conditions hold,

Lξ(i)Lξ( j) gab = 0, (30)

for (i) = (j) and (i) < (j), (i), (j) = 1, 2, .., m or, for (i) = (j) and (i) > (j), (i), (j) = 1, 2, .., m.The remainingLξ(i)Lξ( j) gab �= 0 for (i) > (j)[(i) < (j)] are then determined through (5). In general,

we will demand that none of the vector fields X(i) generate motions.A less demanding definition would be:

Definition 3 (Complete set):A Lie algebra leads to a complete set of weak Lie-symmetries if each of its infinitesimal

operators Xi = ξ a(i)

∂∂xa generates a weak Lie motion: Lξ(i)Lξ(i) gab = 0,Lξ(i) gab �= 0 for every i = 1,

2, . . . , m.

In Sec. V C, examples will be given showing that the alternative Definitions 2 and 3 for completesets of weak Lie symmetries lead to different results. In general, we will prefer Definition 2.

As will be seen in the Appendix a consequence is that if g(X, Y) allows the maximal group ofmotions, weak Lie motions for g(X, Y) do not exist or reduce to conformal motions. As an example:in 2-dimensional Euclidean space with a 3-parameter maximal group (two translations and onerotation), no genuine weak (Lie) motion exists. The other extremal case is the non-existence ofgenuine weak Lie motions, e.g., for the rotation group together with Definition 2. The Kasner metric(11) which allows three space translations as isometries, is a candidate for not leading to genuineweak Lie motions.

V. WEAKLY STATIC AND SPHERICALLY SYMMETRIC METRICS

We now want to determine the metrics allowing a time translation and the rotation group as weakLie motions. The group is chosen such that, as an isometry group, it describes static, sphericallysymmetric (s.s.s.) metrics. Thus we have to allow for four vector fields ξ (i), i = 1, 2, 3, 4 forming aLie algebra with a 2-parameter Abelian subalgebra and then drag twice the arbitrary metric gab. Atfirst, Definition 3 is applied and the target metric γ ab calculated.

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041701-7 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

A. Weakly static metrics

To begin, we demand that only the time translation T = X1 with components ξ s(1) = δs

0 generatesa weak motion: LX1LX1 gab = 0. The resulting class of metrics is

gab = x0cab(x1, x2, x3) + dab(x1, x2, x3), (31)

with arbitrary symmetric tensors cab, dab. The class remains invariant with regard to linear transfor-mations in time x0 → α(x1, x2, x3)x0 + β(x1, x2, x3); α, β arbitrary functions.

B. Weak spherical symmetry

Now, the three generators of spatial rotations SO(3) in a representation using polar coordinatesx1 = r, x2 = θ , x3 = φ are added. Its corresponding generators are

ξ s(2) = δs

3, ξ(3) = −sin x3δs2 − cos x3ctgx2δs

3, ξ(4) = cos x3δs2 − sin x3ctgx2δs

3. (32)

Lie-dragging with the time translation and with ξ (2) forming the Abelian subgroup leads to1γ ab

= gab,0,2γ ab = gab,3, and to the weakly Lie invariant metric (i.e., with LX1LX1 gab = 0,LX2LX2 gab

= 0)

gab = x0x3cab(x1, x2) + x0dab(x1, x2) + x3eab(x1, x2) + fab(x1, x2) (33)

with four arbitrary bilinear forms cab, dab, eab, fab.Lie-dragging with ξ (3) and ξ (4) applied to any of these bilinear forms results in the following

equations (using fab for the presentation):

3γ ab = −sin x3 fab,2 − 2 cos x3 f2(aδ

3b) + 2 sin x3ctgx2 f3(aδ

3b) + 2

cos x3

sin2x2f3(aδ

2b), (34)

4γ ab = cos x3 fab,2 − 2 sin x3 f2(aδ

3b) − 2 cos x3ctgx2 f3(aδ

3b) + 2

sin x3

sin2x2f3(aδ

2b). (35)

The demand2γ ab = 3

γ ab = 4γ ab = 0, i.e., that spherical symmetry holds, leads to cab = eab = 0 and

to the well-known result for fab, dab:

fab = α(x1)δ0aδ

0b − β(x1)δ1

aδ1b − ε(x1)[δ2

aδ2b + sin2x2δ3

aδ3b] (36)

with two free functions α(x1), ε(x1). One of the functions α(x1), β(x1) is superfluous because, locally,a 2-dimensional space is conformally flat. f01 = f23 = 0 follows from the rotation group acting on a2-dimensional subspace. In addition, here f02 = f03 = f12 = f13 = 0 has been used.

If Definition 3 for complete sets of weak symmetry is imposed: two further PDE’s must then besatisfied. If all generators of the rotation group are taken into account, then the result is

γab = x0dab(x1, x2) + fab(x1, x2) (37)

with two bilinear forms dab, fab having the same form:

fab = α(x1)δ0aδ

0b − β(x1)δ1

aδ1b − [x2ε1(x1) + ε2(x1)][δ2

aδ2b + sin2x2δ3

aδ3b]. (38)

For the proof, we do not reproduce here the lengthy full expressions for Lξ(3)Lξ(3) fab = 0 andLξ(4)Lξ(4) fab = 0, but give only the equations for the components f22, f33:

Lξ(3)Lξ(3) f22 = −sin2x2 f22,2,2 + 2cos2x3

sin2x2[− f22 + f33

sin2x2]

!= 0, (39)

Lξ(4)Lξ(4) f22 = −cos2x2 f22,2,2 + 2sin2x3

sin2x2[− f22 + f33

sin2x2]

!= 0. (40)

The consequences f22, 2, 2 = 0 and f33 = sin2x2 f22 are obvious. That (38) is a genuine solution isshown by γ22 = Lξ(3) f22 = −sin x3ε1(x1) �= 0 and by γ33 = Lξ(3) f33 = −sin x3sin2x2ε1(x1) �= 0 ifε1(x1) �= 0.

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041701-8 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

The surface x1 = const, x0 = const has Gaussian curvature:

K = 1

2(ε1x2 + ε2)2[−ε1ctgx2 + 2ε1x2 + 2ε2 + (ε1)2

ε1x2 + ε2]. (41)

ε1, ε2 are now constants. For ε1 → 0 we obtain the constant curvature of the 2-sphere.The time translation and the 3 generators of the rotation group form a complete set of weak Lie

motions; this shows that Definition 3 is not empty.However, if it is asked that the rotation group generate a strong set of weak symmetries according

to Definition 2, then the result is very restrictive. The conditions Lξ(2)Lξ(3) fab = 0 = Lξ(2)Lξ(4) fab forEqs. (33) and (38) are leading to the remaining metric tensor of (37). If Lξ(3)Lξ(4)γ33 = 0 is studiedfor fab, then Lξ(3)Lξ(4) fab �= 0 due to the only nonvanishing expression Lξ(3)Lξ(4) f33 = sin x2cos x2

× ε1(x1) for ε1(x1) �= 0. Thus the demand that the rotation group in 3 dimensions generatesa strong set of weak Lie symmetries according to Definition 2 enforces ε1(x1) = 0 and re-duces to an isometry. Nevertheless, the resulting spherically symmetric metric is only weaklystatic.

C. The group G3 acting as a group of weak Lie motions

In taking up the example of a G3 acting on V3 from Sec. II with Lie algebra (18), we first applyDefinition 3 to a scalar f(x1, x2, x3). If the generators are to lead to motions, then the only solution isf = constant. We find

LX1LX1 = f,2,2 = 0,LX2LX2 = f,3,3 = 0, (42)

LX3LX3 = f,1,1 − 2x3 f,2,1 + (x3)2 f,2,2 = 0,LX1LX2 = LX2LX1 = f,2,3 = 0, (43)

LX1LX3 = − f,1,2 + x3 f,2,2 = 0,LX3LX1 = − f,1,2 + x3 f,2,2 = 0, (44)

LX2LX3 = − f,1,3 + f,2 + x3 f,2,3 = 0,LX3LX2 = − f,3,1 + x3 f,3,2 = 0. (45)

With these results, Definition 3 for a complete set of weak Lie motions leads to

f = a0x2x3 + b0x1(x2 − x1x3) + c0x1x3 + b1x2 + c1x3 + d1x1 + d0, (46)

while Definition 2 results in

f = c0(x1x3 + x2) + c1x3 + d1x1 + d0. (47)

We note, that the only one of the 9 possible demands so far unused, i.e., Lξ(3)Lξ(2) gab = 0 reduces(47) to

f = c1x3 + d1x1 + d0. (48)

Applying G3 to the metric, the following weakly Lie-invariant metric is obtained,

γab =

x2

⎛⎜⎝

(0)α 11

(0)α 12 P1

(0)α 21

(0)α 22 P ′

1

P1 P ′1 P2

⎞⎟⎠ + x3[x1

⎛⎜⎝

(0)α 11

(0)α 12 P1

(0)α 21

(0)α 22 P ′

1

P1 P ′1 P2

⎞⎟⎠ +

⎛⎜⎜⎝

(0)β 11

(0)β 12 P1

(0)β 21

(0)β 22 P ′

1

P1 P ′1 P2

⎞⎟⎟⎠] +

⎛⎝

Q1 Q′1 Q2

Q′1 Q1 Q′

2

Q2 Q′2 Q3

⎞⎠,

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041701-9 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

where Pi , P ′i , Qi , Qi , Q′

i are polynomials in the coordinate x1 of order i, the coefficients of whichare not all independent:

P1 = (0)α 12x1 + c13, P ′

1 = (0)α 22x1 + c23, P2 = (0)

α 22(x1)2 + 2c23x1 + c33,

Q1 =(0)l 11x1 + (0)

m11, Q′1 =

(0)l 12x1 + (0)

m12, Q1 = l22x1 + (0)m22,

Q2 =(0)l 12(x1)2 + (0)

m13x1 +(0)k 13, Q′

2 =(0)l 22(x1)2 + (0)

m23x1 +(0)k 23,

Q3 =(0)l 22(x1)3 + (0)

m23(x1)2 +(0)k 33x1 + m33 and

(0)α ab, (a, b = 1, 2), ci j ,

(0)l i j ,

(0)mi j , and

(0)k i j constants.

In the polynomials P1, P ′1, P2, the constants αab, cab are exchanged by the set of indepen-

dent constants βab, dab. Thus Definition 2 is not empty. Two independent matrices of thetype that occurred for the group acting as an isometry group and a third, new matrix occurnow.

Definition 3 leads to a different complete set of weak Lie motions for which the metrictakes the form: gab = dab(x1)x2 + eab(x1)x3 + εab(x1) with d, e, ε expressed by matrices of theform:

⎛⎜⎝

P11 P12 Q1

P12 P22 Q2

Q1 Q2 M

⎞⎟⎠,

where Pik are polynomials of 1st degree, Qi of 2nd degree, and M a polynomial of 3rd degree.

VI. A NEW ALGEBRA STRUCTURE

For Lie-dragging, up to now we have mostly taken vector fields forming Lie algebras corre-sponding to Lie groups of point transformations. In the following, we consider more general typesboth of groups and algebras in Secs. VII and VIII.

A. Lie-dragging for vector fields not forming Lie algebras

Already in (25) of Sec. IV A, vector fields containing free functions were considered. Wenow continue with vector fields X1 = ξ r ∂

∂xr ; X2 = ηs ∂∂xs with ξ r = f (x0)δr

1, ηs = h(x1)δr

0 suchthat

[X1, X2] = f (x0)H (x1)X2 − h(x1)F(x0)X1. (49)

Here, F(x0) = d(ln f (x0))dx0 , H (x1) = d(lnh(x1))

dx1 . The finite transformations belonging to X1 and X2, re-spectively, are generalized time- and space-translations

x0 → x0′ = x0 + h(x1); x1 → x1′ = x1 + f (x0) (50)

leaving invariant the time interval |x0(i) − x0

( j)| and the space interval |x1(i) − x1

( j)| between two events(x0

(i), x1(i)) and (x0

( j), x1( j)). Each of the two families of transformations

x0 → x0′ = x0 + h(x1); x1 → x1′ = x1 + a, (51)

and

x1 → x1′ = x1 + f (x0); x0 → x0′ = x0 + b (52)

forms a group. However, these groups are not Lie groups: in part, the Lie-group parameters havebeen replaced by arbitrary functions. In this case, the algebra (49) reduces to either

[X1, X2] = H (x1)X2, (53)

or to

[X1, X2] = −F(x0)X1. (54)

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041701-10 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

Likewise, (49), (53), and (54) are not Lie algebras. Both transformations (50) applied together donot even form a group.26

Equation (52) is a subgroup of the so-called Mach-Poincare group27 G4(3) (cf. also Ref. 28),pp. 85–101):

xa′ = Aar xr + f a(x0), x0′ = x0 + b, Aa

r Arb = δa

b . (55)

This group plays a role in Galilean relative mechanics.A generalization is the group G1(6) of transformations leaving invariant the observables de-

scribing a rigid body; 6 free functions of x0 and one Lie-group parameter do appear,

xi ′ = Aij (x

0)x j + f i (x0), x0′ = x0 + b, Aij (x

0)A jk (x0) = δi

k, (i, j = 1, 2, 3). (56)

The corresponding seven algebra generators are

T = ∂

∂x0, Xi = fi (x

0)∂

∂xi(i not summed),

Y1 = ω23(x3 ∂

∂x2− x2 ∂

∂x3), Y2 = ω1

3(x3 ∂

∂x1− x1 ∂

∂x3), Y3 = ω1

2(x2 ∂

∂x1− x1 ∂

∂x2)

(57)

with ωij = ωi

j (x0). The corresponding algebra is given by

[T, T ] = 0, [T, Xi ] = Fi (x0)Xi , Fi = d

dx0ln( fi (x

0)), [Xi , X j ] = 0, (i, j = 1, 2, 3)

[T, Y1] = �23(x0)Y1, [T, Y2] = �3

1(x0)Y2, [T, Y3] = �12(x0)Y3,�

ij = d

dx0ln(ωi

j (x0)),

[Y1, Y2] = −ω13ω

23

ω12

Y3, [Y2, Y3] = −ω12ω

13

ω23

Y1, [Y1, Y3] = −ω12ω

32

ω13

Y2,

[X1, Y1] = 0, [X1, Y2] = f1(x0)

f3(x0)ω1

3 X3, [X1, Y3] = − f1(x0)

f2(x0)ω1

2 X2,

[X2, Y1] = f2(x0)

f3(x0)ω2

3 X3, [X2, Y2] = 0, [X2, Y3] = f2(x0)

f1(x0)ω1

2 X1,

[X3, Y1] = f3(x0)

f2(x0)ω2

3 X2, [X3, Y2] = − f3(x0)

f1(x0)ω1

3 X1, [X3, Y3] = 0. (58)

There exist further groups of this non-Lie type occurring in the classical relative mechanics likeWeyl’s kinematical group G3(6) and the covariance group of the Hamilton-Jacobi equation G7(3)or, as a subgroup in non-relativistic quantum mechanics, the covariance group of the Schrodingerequation28 G12(0). The structure functions of all these groups depend on a single coordinate, thetime.

VII. EXTENDED LIE ALGEBRAS

In the following, we will deal with a subbundle of the tangent bundle of n-dimensional Euclideanor Lorentz space. We will permit that the structure constants in the defining relations for a Lie algebrabecome dependent on the components ξ a

i of the vector fields Xi(x): they will become structurefunctions.

Definition 4: The Lie brackets

[Xi , X j ] = cki j (x

1, x2, . . . , r )Xk (59)

with structure functions cki j (x

1, x2, . . . , xr ) are said to build an extended Lie algebra.

The Lie algebra elements form an “involutive distribution.” This is a smooth distribution V ona smooth manifold M, i.e., a smooth vector subbundle of the tangent bundle TM. The Lie brackets

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041701-11 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

constitute the composition law; the injection V ↪→ T M functions as the anchor map.29 This isa simple example for a tangent Lie algebroid (cf. also Ref. 3, p. 100 and example 2.7, p. 105).Nevertheless, the involutive distribution used here can also be considered a subset of the infinite-dimensional “Lie-algebra”B(M). Closely related, but different structures are a family of Lie algebras(Refs. 30 and 31) and variable Lie algebras (Ref. 32, p. 115).

After completion of the paper, I learned of some of the historical background of (59): It alreadyhas occurred as the condition for closure of a complete set of linear, homogeneous operators belong-ing to a complete system of 1st order PDE’s in Jacobi’s famous paper of 1862 (Ref. 33, §26, p. 40).In Jacobi’s paper, (59) is used in phase space such that the structure functions depend on bothcoordinates and momenta: ck

i j (x1, x2, . . . , xr , p1, p2, ..pr ). It is in Clebsch’s paper of 1866

(Ref. 34, §1) in connection with his definition of a complete system of linear PDE’s that therhs of (59) depends only on the coordinates. Cf. also Eq. (3.1) in Ref. 2., p. 311.

If an extended Lie algebra is studied instead of an ordinary Lie algebra, (7) must be replaced by

LXiLX j Xk = (cljkcm

il + Xi cmjk)Xm, (60)

and (8) by

cljkcm

il + cli j c

mkl + cl

ki cmjl + Xi c

mjk + Xkcm

i j + X j cmki = 0. (61)

An extended Cartan-Killing form can be defined acting as a symmetric metric on the sections of thesubtangent bundle. An asymmetric form could be defined as well.

Definition 5 (Generalized Cartan-Killing form):The generalized Cartan-Killing bilinear form τ is defined by

τi j := σi j + 2X (i cmj)m = cm

il cljm + 2X (i c

mj)m . (62)

The generalized Cartan-Killing form now depends on the base points of the fibres in the tangentbundle. They may be interpreted as a metric.

We use the example of the group G1(6) given in Sec. VI A. If we use the notation Z1 = T,Zi = Xi, (i = 1, 2, 3), Zj = Yj(j = 1, 2, 3) where Xi correspond to time-dependent translations, Yj

to time-dependent rotations, the structure functions for the corresponding extended algebra (58) ofrigid body transformations are given by

ci0i = (ln fi )

˙ (i = 1, 2, 3), c404 = (lnω2

3)˙, c505 = (lnω1

3)˙, c606 = (lnω1

2)˙

cAi j = 0 (i, j = 1, 2, 3, A = 0, .., 6), c6

45 = −ω23ω

13

ω12

, c456 = −ω1

3ω12

ω32

c546 = −ω3

2ω12

ω31

cA14 = 0 (A = 0, .., 6), c3

15 = − f1

f3ω1

3, c216 = − f1

f2ω1

2

c324 = − f2

f3ω2

3, cA25 = 0 (A = 0, .., 6), c1

26 = f2

f1ω1

2,

c234 = f3

f2ω2

3, c135 = f3

f1ω1

3, cA36 = 0 (A = 0, .., 6).

From them, calculation of the extended Cartan-Killing form leads to a Lorentz metric withsignature (1,3) of rank 4 within a degenerated 7-dimensional bilinear form,

τi j =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

τ00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 τ44 0 00 0 0 0 0 τ55 00 0 0 0 0 0 τ66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (63)

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041701-12 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

where τ00 = �3i=1

fi

fi+ ω2

3

ω23+ ω3

1

ω31+ ω1

2

ω12, and τ44 = −4(ω2

3)2, τ55 = −4(ω13)2, τ66 = −4(ω1

2)2. By pro-

jection into the 4-dimensional space with coordinates 0, 4, 5, 6 and signature (1,3), we surprisinglyarrive at the general class of one-dimensional gravitational fields (Ref. 26). For special values forthe fi, and ωk

i , the Kasner metric (11) can be derived by this approach. All the pre-relativistic groupsmentioned at the end of Sec. VI A lead to Cartan-Killing forms depending on just one coordinate,the time.

The following definition introduces a new class of extended motions and a new class of weakextended motions, the infinitesimal generators of which form an extended Lie algebra.

Definition 6 (Extended motions):Let x → x + ξ , y → y + η be infinitesimal transformations forming a continuous group the

corresponding algebra of which is an extended Lie algebra according to Definition 4. Then, thevector fields X = ξ c ∂

∂xc , Y = ηc ∂∂xc with LX gab = 0,LY gab = 0 are called extended motions.

An analogous formulation is:

Definition 7 (Extended weak motions):Let x → x + ξ , y → y + η be infinitesimal transformations forming a continuous group

the corresponding algebra of which is an extended Lie algebra according to Definition 4. Then,the vector fields X = ξ c ∂

∂xc , Y = ηc ∂∂xc with LXLX gab = 0,LYLY gab = 0 are called extended weak

motions.

VIII. EXTENDED MOTIONS AND EXTENDED WEAK (LIE) MOTIONS

In Sec. VI A, we have given examples of non-Lie groups leading to extended Lie algebras. Howwill the corresponding extended motions and extended weak (Lie) motions differ? These conceptsare exemplified here with the most simple non-Lie group (52). The tangent vectors X, Y with thealgebra (54) form an extended motion (LX gab = 0,LY gab = 0) for all metrics of maximal rank 3:

gab =

⎛⎜⎜⎝

α00(x2, x3) 0 α02(x2, x3) α03(x2, x3)0 0 0 0

α02(x2, x3) 0 α22(x2, x3) α23(x2, x3)α03(x2, x3) 0 α23(x2, x3) α33(x2, x3)

⎞⎟⎟⎠, (64)

with arbitrary functions αab due to arbitrariness of f(x0). This is to be compared with the motionsderived from X = ∂

∂x1 , Y = ∂∂x0 forming an Abelian Lie algebra and leading to

gab = αab(x2, x3). (65)

The corresponding extended weak (Lie) motions (LXLX gab = 0,LYLY gab = 0) are given by

gab =

⎛⎜⎜⎝

x1α00 + β00 β01 x1α02 + β02 x1α03 + β03

β01 0 β12 β13

x1α02 + β02 β12 x1α22 + β22 x1α23 + β23

x1α03 + β03 β13 x1α23 + β23 x1α33 + β33

⎞⎟⎟⎠, (66)

where αab = αab(x2, x3); βab = βab(x2, x3), α01 = 0, β11 = 0. Comparison with the weak (Lie)motions generated by the translations given above shows the class of metrics:

gab = x1αab(x2, x3) + βab(x2, x3). (67)

IX. DISCUSSION AND CONCLUSION

When Lie-dragging is seen as a mapping in the space of metrics, it may be asked whether it couldprovide a method for generating solutions of Einstein’s equations from known solutions. It is easily

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041701-13 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

shown that the Schwarzschild vacuum solution, the Robertson-Walker metric with flat 3-spaces, andthe Kasner metric cannot be obtained by Lie-dragging of Minkowski space. On the other hand, themetric (33) which is weakly Lie-invariant with respect to the group (T, SO(3)) trivially containscosmological solutions of Einsteins equation. If the metric x0dab(x1, x2) with spherically symmetryand with flat space sections is chosen, by a transformation of the time coordinate we arrive at theline element

ds2 = (dτ )2 − 2/3τ 3/2[(dr )2 + r2(dθ )2 + r2sin2 θ (dφ)2]. (68)

It describes a cosmic substrate with the equation of state p = − 19μ, where p describes pressure and

μ the energy density of the material. This equation of state for w = − 19 is non-phantom because of

−1 < w but does not accelerate the expansion of the universe which occurs for −1 < w < − 13 .

It remains to be seen whether the anisotropic line element

ds2 = (dτ )2 − c0τ2/3[c1θr + c2][(dr )2 + r2(dθ )2 + r2sin2 θ (dφ)2] (69)

can satisfy Einstein’s equations with a reasonable matter distribution. In view of the fact thatLie-dragging does not preserve the rank of the metric, its efficiency for generating interpretablegravitational fields is reduced considerably.

Surprisingly, by studying the rigid body transformations G1(6) as a group of extended motions,we arrived at the complete class of one-dimensional gravitational fields including the Kasner metric.More generally, a close relation to finite transformation groups in classical, non-relativistic mechanicscontaining arbitrary functions has been established.

A classification of solutions of Einstein’s equations with regard to weak (Lie) symmetries couldbe made. Although this might be a further help for deciding whether two solutions are transformableinto each other or not, the calculational effort looks extensive.

Weak Lie-invariance as a weakened concept of “symmetry” has been introduced and its conse-quences presented through a number of examples. It also has led to the introduction of a new typeof algebra (“extended Lie algebra”) which is an example for a tangent Lie algebroid. In each fibreof a subbundle of the tangent bundle, the “extended Lie algebra” reduces to a Lie algebra. By helpof an extended Cartan-Killing form, Riemann or Lorentz metrics have been constructed on such analgebroid. A particular example is provided by the non-Lie groups of classical mechanics mentionedabove.

A classification of non-Lie groups leading to extended Lie algebras and of the extended Liealgebras in n-dimensional space arises. A preliminary study for the case n = 2 has shown, that thisis a delicate problem depending on the existence of solutions of some 1st-order nonlinear PDEs .

Whether there are noteworthy applications in geometry and physics beyond those establishedhere for classical mechanics and the Schrodinger equation will have to be found out.

ACKNOWLEDGMENTS

My sincere thanks go to A. Papadopoulos for inviting me to the Strasbourg conference atIRMA. Remarks by P. Cartier and Y. Kosmann-Schwarzbach, participants at the conference, werequite helpful. I am also grateful for advice on mathematical concepts like algebroids by H. Seppanenand Ch. Zhu of the Mathematical Institute of the University of Gottingen as well as to my colleagueF. Muller-Hoissen, Max Planck Institute for Dynamics and Self-Organization, Gottingen, for aclarifying conceptual discussion. As a historian of mathematics, E. Scholz, University of Wuppertal,did guide me to the relevant historical literature. The presentation did also profit much from thesuggestions of a referee.

APPENDIX: INTEGRABILITY CONDITIONS

That an arbitrary symmetrical tensor field of fixed rank cannot be reached by the operationof Lie-dragging may be seen already from (15) and (13), or from the following equations obtained

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041701-14 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

from (16):g∇b

g∇cξa + ξd Rd

bca(g) = 1/2[g∇bγac +

g∇cγba +

g∇aγcb]. (A1)

Here, Rdbca(g) is the curvature tensor of the metric gab. For gab, γ ab fixed, the n3 equations (A1) would

be an integrability condition for the n components of the vector field ξ . Equation (A1) generalizespart of the integrability condition for (15) given in Ref. 15, Eq. (6.2), p. 56. As an example, we takeMinkowski space gab = ηab, a space of maximal symmetry. From (A1) with γ ab = 2ξ (a, b) follows:

∂b∂cξa = ∂bξ(a,c) + ∂cξ(b,a) + ∂aξ(c,b), (A2)

the general solution of which, apart from the generators of the Poincare group, is ξ a = c0ηarxr

+ ∂rF[rs]ηas. Thus, a homothetic motion appears as well.If we look at Eq. (A1) as a condition for γ ab when the vector field ξ and the metric gab are given,

the equation then says that there are linear relations between the first derivatives of γ ab. Furtherdifferentiation of (A1) leads to

g∇d

g∇b

g∇cξa +

g∇a

g∇b

g∇dξc +

g∇c

g∇b

g∇aξd = 1/2[

g∇d

g∇bγac +

g∇a

g∇bγcd +

g∇c

g∇bγda]

+γas Rsbcd (g) + γcs Rs

bda(g) + γds Rsbac(g) = 0. (A3)

A counting of derivatives and equations leads to the number of restrictions for the obtainable γ ab

showing up explicitly as relations among the derivatives of γ ab.From (A2), by contraction with ηbc the equation ∂a(∂cξ c) = 0 follows, whence ∂cξ c = c0

= const. Contraction with ηac then leads to (∇)2ξ c = 0. The most general ansatz for solving ∂cξ c

= c0 is ξ c = c04 xc + ∂r F [rc] with ∂r(∇)2F[rc] = 0. Let Xc := ∂rF[rc]; then, from (A2) ∂c(∂aXb

+ ∂bXa) = 0. Whence follow the equation for a homothetic motion.

1 D. E. Rowe, “The early geometrical works of Sophus Lie and Felix Klein,” in The History of Modern Mathematics, Vol. I,Ideas and Their Reception, edited by David E. Rowe and John McCLeary (Academic Press, Boston, 1989), pp. 209–273.

2 T. Hawkins, “Line geometry, differential equations and the birth of Lie’s theory of groups,” in The History of ModernMathematics, Vol. I, Ideas and Their Reception, edited by David E. Rowe and John McCLeary (Academic Press, Boston,1989), pp. 275–327.

3 K. C. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids (Cambridge University Press, Cambridge, UK,2005).

4 P. Libermann, “Lie algebroids and mechanics,” Arch. Math. (Brno) 32, 147–162 (1996).5 E. Martınez, “Lagrangian mechanics in Lie algebroids,” Acta Appl. Math. 67, 295–320 (2001).6 K. Grabowska, P. Urbanski, and J. Grabowski, “Geometrical mechanics on algebroids,” Int. J. Geom. Methods Mod. Phys.

3, 559–575 (2006).7 Latin indices from the beginning (a, b, c, ..) and end of the alphabet (r, s, t, ..) run from 1 to n or 0 to n-1 where n is the

dimension of the space considered. Indices from the middle (i, j, k, l, ..) may take other values. The summation conventionis used except when indicated otherwise.

8 Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translation of MathematicalMonographs, Vol. 182, edited by I. S. Krasil’shchik and A. M. Winogradov (American Mathematical Society, Providence,1997).

9 In current mathematical literature, the definition of a Lie algebra is much more general. It is defined either as a moduleB(M) of the set of all C∞-vector fields on a C∞-manifold with a multiplication introduced via the Lie-bracket, or as afinite-dimensional vector space V over the real or complex numbers with a bilinear multiplication on it defined by ananti-commuting bracket [, ] satisfying the Jacobi identity (4).

10 B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, UK, 1982).11 Symmetrization brackets are used: A(r Bs) = 1

2 (Ar Bs + As Br ); A[r Bs] = 12 (Ar Bs − As Br ).

12 For mechanical systems in phase space, this infinitesimal symmetry transformation is applied to the generalized coordinatesand supplemented by an infinitesimal transformation for the momenta: pa → pa′ = pa + ηa with an additional infinitesimalgenerator ηa. Some authors use the name “weak-Lie” symmetry for what we would name Lie symmetry; cf. T. Pang, J.-H.Fang, P. Lin, M.-J. Zhang, and K. Lu, “New type of conserved quantities of Lie symmetry for nonholonomic mechanicalsystems in phase space,” Commun. Theor. Phys. (Beijing) 52, 977–980 (2009).

13 A. Z. Petrov, Einstein Spaces (Pergamon Press, Oxford, 1969), p. 213.14 K. Yano, The Theory of Lie Derivatives and its Applications (North-Holland Publ. Co., Amsterdam, 1957).15 R. Maartens, “Affine collineations in Robertson-Walker space time,” J. Math. Phys. 28, 2051 (1987).16 G. H. Katzin, J. Levine, and W. R. Davis, “Curvature collineations: A fundamental symmetry property of the space-times

of general relativity. . . .,” J. Math. Phys. 10(4), 617–629 (1969).17 C. D. Collinson, “Conservation laws in general relativity based upon the existence of preferred collineations,” Gen. Relativ.

Grav. 1(2), 137–142 (1970).18 A. D. Papadopoulos, “P-invariance in general relativity,” Tensor N. S. 40, 135–143 (1983).

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041701-15 Hubert Goenner J. Math. Phys. 54, 041701 (2013)

19 H. Goenner, “Killing vector Fields and the Einstein-Maxwell field equation for null electromagnetic fields,” Gen. Relativ.Grav. 16, 795–796 (1984).

20 G. Fubini, “Sugli spazii a quattro dimensioni che ammettono un gruppo continuo di movimenti,” Annalen der Mathematik[Germany] 9, 33–90 (1903).

21 I. P. Yegorov, “Motions in spaces of affine connectivity,” Ph.D. dissertation (Moscow State University, 1955), p. 202.Quoted from Ref. 14, p. 134.

22 G. Yu. Bogoslovsky, “A special relativistic theory of the locally anisotropic space-time. I. The metric and group of motionsof the anisotropic space of events. II. Mechanics and Electrodynamic in the anisotropic space,” Nuovo Cimento B 40,99–115 (1977); 40, 116–134 (1977).

23 G. Yu. Bogoslovsky, “A viable model of locally anisotropic space-time and the Finslerian generalization of the relativitytheory,” Fortschr. Phys. 42, 143–193 (1994).

24 G. Yu. Bogoslovsky and H. F. Goenner, “Finslerian spaces possessing local relativistic symmetry,” Gen. Relativ. Grav. 31,1565–1603 (1999).

25 H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nded. (Cambridge University Press, Cambridge, UK, 2003).

26 The class of functions involved may be narrowed considerably by the demand that the function f of a special type be keptfixed, e.g., be a polynomial of degree p, or f (x0) = a sin x0 + b cos x0. In these cases, just one function with constantcoefficients occurs in the group; the group transformations change only the coefficients.

27 H.-J. Treder, Die Relativitat der Tragheit (Akademie Verlag, Berlin, 1972).28 H. Goenner, “Machsches Prinzip und Theorien der Gravitation,” in Grundlagenprobleme der modernen Physik, edited by

J. Nitsch, J. Pfarr, and E.-W. Stachow (B.I.-Wissenschaftsverlag, Mannheim, 1981).29 C.-M. Marle, “Calculus on Lie algebroids, Lie groupoids and Poisson manifolds,” Diss. Math. 457, 1–57 (2008); here

p. 13.30 A. Douady and M. Lazard, “Espaces fibres en algebres de Lie et en groupes,” Invent. Math. 1, 133–151 (1966).31 D. Coppersmith, “A family of Lie algebras not extendible to a family of Lie groups,” Proc. Am. Math. Soc. 66, 365–366

(1977).32 H. Leptin and J. Ludwig, Unitary Representation Theory of Exponential Lie groups, De Gruyter Expositions in Mathematics

(W. de Gruyter, Berlin/New York, 1994), Vol. 18.33 C. Jacobi, “Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium quemcunque

propositas integrandi,” J. Reine Angew. Math. 60, 1–181 (1862).34 A. Clebsch, “Uber die simultane Integration linearer partieller Differentialgleichungen,” J. Reine Angew. Math. 65, 257–

268 (1866).

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