Lie algebraic CarrollGalilei duality

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arXiv:2210.13924v1 [math.DG] 25 Oct 2022

LIE ALGEBRAIC CARROLL/GALILEI DUALITY

JOSÉ FIGUEROA-O’FARRILL

În memoria Veronicăi Stanciu

Abstract. We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures.

Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or ga-

lilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric

derivation. This is the same data deﬁning a one-dimensional double extension of the metric Lie algebra

and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie

algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspondence

between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras. This reformulation

allows us to use the structure theory of metric Lie algebras to give a list of bargmannian, carrollian

and galilean Lie algebras in the positive-semideﬁnite case. We also characterise Lie groups admitting a

bi-invariant (ambient) leibnizian structure. Leibnizian Lie algebras extend the class of bargmannian Lie

algebras and also set up a non-canonical correspondence between carrollian and galilean Lie algebras.

Contents

1. Introduction 1

2. Bargmannian, carrollian and galilean structures 5

3. Bargmannian, carrollian and galilean Lie algebras 5

3.1. Metric Lie algebras and double extensions 5

3.2. Bargmannian Lie algebras 6

3.3. Carrollian Lie algebras 6

3.4. Galilean Lie algebras 7

3.5. Summary 8

4. Classiﬁcation 8

5. Leibnizian Lie groups 10

5.1. Leibnizian structures 10

5.2. Leibnizian Lie algebras 10

5.3. A leibnizian Lie algebra which is not bargmannian 12

6. Conclusion 13

Acknowledgments 13

Data availability 13

Appendix A. Skew-symmetric derivations of a reductive metric Lie algebra 13

Appendix B. Once more, with indices 14

References 15

1. Introduction

The study of non-lorentzian spacetime geometries is coming of age (see, e.g., the recent reviews [1,2]),

yet there are still some simple questions which have not been asked nor answered. A natural ﬁrst step

when studying an unfamiliar geometric structure is to ﬁnd examples of such structures with lots of

symmetries. A natural class of such examples are homogeneous spaces and, in particular, Lie groups

with bi-invariant structures. Bi-invariance is typically quite strong and this often allows one to classify

them or at least to characterise them in linear algebraic terms. This is the case, for example, with

Lie groups admitting a bi-invariant metric (of any signature), which were studied by Medina [3] and

characterised in terms of their Lie algebras by Medina and Revoy [4] (see also [5–7]).

EMPG-22-20, ORCID: 0000-0002-9308-9360.

2 JOSÉ FIGUEROA-O’FARRILL

The three protagonists of today’s tale are Bargmann, Carroll and Galilei. They may be used to

label Lie groups, Lie algebras, homogeneous spaces and also Cartan geometries (here, equivalently, G-

structures). It turns out that in all of these settings, objects with these names sit in relation to each other

in a way which suggests a correspondence (loosely, a duality) between Carroll and Galilei mediated by

Bargmann. This correspondence was ﬁrst pointed out in a geometric context in [8], but before describing

this result let us set the stage by discussing the Lie algebras themselves.

The Carroll and Galilei Lie algebras are examples of kinematical Lie algebras [9,10]. In spatial

dimension n, they are spanned by (Lab,Ba,Pa,H), where Lab = −Lba span a Lie subalgebra isomorphic

to so(n)under which Ba,Paare vectors and Ha scalar. These conditions translate into the following Lie

brackets which are common to all kinematical Lie algebras

[Lab,Lcd] = δbcLad −δacLbd −δbdLac +δbdLac

[Lab,Bb] = δbcBa−δacBb

[Lab,Pb] = δbcPa−δacPb

[Lab,H] = 0.

(1.1)

The kinematical Lie algebra where all other brackets vanish is called the static kinematical Lie algebra

and denoted s. All kinematical Lie algebras are deformations of s[11–13]. The subalgebra s0of sspanned

by (Lab,Ba,Pa)will play a rôle in our discussion.

The Carroll Lie algebra cis a central extension of s0, with Hthe central element and additional

nonzero bracket

[Ba,Pb] = δabH. (1.2)

In contrast, the Galilei Lie algebra gis an “extension-by-derivation” of s0. The derivation is adH= [H,−]

and is deﬁned by adH(Lab) = adH(Pa) = 0 and adH(Ba) = −Pa, resulting in the additional nonzero

bracket

[H,Ba] = −Pa. (1.3)

We may summarise these observations in the diagrammatical language of (short) exact sequences of

Lie algebras as follows:

0s0gR0

(1.4)

The exact row says that gis an extension-by-derivation of s0, whereas the exact column says that cis a

one-dimensional extension of s0. The diagram does not ﬁx the brackets uniquely, since s0admits many

derivations and also other one-dimensional extensions, central or not.

The Bargmann Lie algebra bis a central extension of the Galilei Lie algebra gwith additional generator

Mand additional bracket

[Ba,Pb] = δabM, (1.5)

which is reminiscent of the bracket (1.2) with Mplaying the rôle of H. Under this relabelling of bases,

we see that the Bargmann Lie algebra bis an extension-by-derivation of the Carroll Lie algebra c. The

derivation is adH= [H,−] where again adHannihilates Lab,Pa,Mand its action on Bais given by the

bracket (1.3).

LIE ALGEBRAIC CARROLL/GALILEI DUALITY 3

This allows us to complete the diagram (1.4) to the following commutative diagram of Lie algebras:

0 0

R R

0c b R0

0s0gR0

0 0

(1.6)

which will recur in other contexts in this work with other Lie algebras playing the rôles of s0,c,gand

b. In fact, this very diagram has already appeared in [14, App. B] in the context of the hamiltonian

description of particle dynamics on the (spatially isotropic) homogeneous galilean spacetimes classiﬁed

in [15]. In that context, the Lie algebras which play the rôles of s0,c,gand bare inﬁnite-dimensional.

The two exact rows in the the diagram (1.6) say that b(resp. g) is an extension-by-derivation of c(resp.

s0), whereas the two exact columns say that b(resp. c) is a one-dimensional extension of g(resp. s0).

We could say, borrowing the terminology from the theory of metric Lie algebras, that the Bargmann Lie

algebra bis a double extension1of s0. This is more than a mere analogy and we will see that this is

exactly right for Lie algebras admitting ad-invariant bargmannian structures (to be deﬁned below).

The Bargmann Lie algebra bcan also be deﬁned as the subalgebra of the Poincaré Lie algebra

iso(n+1, 1)in one dimension higher which centralises a null translation. Since the Poincaré transla-

tions commute, they are contained in band hence the (simply-connected) Bargmann Lie group acts

transitively on Minkowski spacetime Mn+2. This fact underlies the geometric Carroll/Galilei duality

in [8].

Choosing a Witt frame (ea,e+,e−)for Minkowski spacetime Mn+2we may express the generat-

ors of the Poincaré algebra as (Lab,L+a,L−a,L+−,Pa,P+,P−). The centraliser of P+is spanned by

(Lab,L+a,Pa,P+,P−)and it is isomorphic to the Bargmann Lie algebra b, with P+playing the rôle of M.

If Xis an element of the Poincaré Lie algebra, we let ξXdenote the corresponding Killing vector ﬁeld on

Minkowski spacetime. The null vector ﬁeld ξP+on Minkowski spacetime is not just Killing but actually

parallel. It deﬁnes a distribution ξ⊥

P+⊂TMwhich is integrable. The leaves of the corresponding foliation

are null hypersurfaces and are copies of the Carroll spacetime Cn+1. Indeed, they are homogeneous

spaces of the normal subgroup with Lie algebra the ideal of bspanned by (Lab,L+a,Pa,P+), which is

isomorphic to the Carroll Lie algebra c, with P+playing the rôle of H. On the other hand, the null re-

duction [16,17] of Mby the one-parameter subgroup generated by ξP+is isomorphic to Galilei spacetime

Gn+1. Indeed, the quotient is homogeneous under the Lie group whose Lie algebra is the quotient of b

by the line spanned by P+, which is isomorphic to the Galilei Lie algebra g. This results in the following

suggestive diagram

Cn+1Mn+2Gn+1,(1.7)

which is the fundamental example of the geometric Carroll/Galilei duality in [8].

This duality seems to be broken when we take all (spatially isotropic) homogeneous kinematical

spacetimes into consideration. In the classiﬁcation of [15] there are three other homogeneous carrollian

spacetimes besides the Carroll spacetime: the carrollian limits dSC of de Sitter and AdSC of anti de Sitter

spacetimes, and the lightcone LC, which can be realised as null hypersurfaces in de Sitter, anti de Sitter

and Minkowski spacetimes, respectively [15,18]. On the other hand, there are two one-parameter families

of homogeneous galilean spacetimes. The galilean limit dSG of de Sitter spacetime is one point (γ= −1)

in a continuum dSGγ, for γ∈[−1, 1], of galilean spacetimes. Similarly, the galilean limit AdSG of

anti de Sitter spacetime is one point (χ=0)in a continuum AdSGχ, for χ>0, of galilean spacetimes.

These two continua have a point in common, since limγ→1dSGγ=limχ→∞AdSGχ. It is often convenient

to describe homogeneous spaces inﬁnitesimally via their Klein pairs. The above homogeneous galilean

1To be clear, this proposal expands the deﬁnition of a double-extension, transcending its origin in the context of metric

Lie algebras, to a more general notion in which a double extension is the composition of a one-dimensional extension with

an extension-by-derivation; at least in those cases where these two operations commute.

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arXiv:2210.13924v1[math.DG]25Oct2022LIEALGEBRAICCARROLL/GALILEIDUALITYJOSÉFIGUEROA-O’FARRILLÎnmemoriaVeronicăiStanciuAbstract.WecharacteriseLiegroupswithbi-invariantbargmannian,galileanorcarrollianstructures.Localisingattheidentity,weshowthatLiealgebraswithad-invariantbargmannian,carrollianorga-lile...

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Lie algebraic CarrollGalilei duality

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