Coisotropy-one homogeneous spacetimes I: (3,2)-Kinematical Lie algebras with $d\geq 3$

Introduction

More than half a century ago, Bacry and Lévy-Leblond [1] started what could be termed the kinematical Erlanger programme, in effect adding “time” to Klein’s famous programme of studying geometries via their symmetry groups. In their original paper they classified kinematical Lie algebras for a class of four-dimensional spacetimes. The corresponding kinematical Lie groups act on the spacetimes as a composition of rotations, boosts and spatial and temporal translations. Their classification was subject to the assumptions of spatial isotropy and the existence of time-reversal and parity automorphisms. These latter “by no means compelling” [1] assumptions were lifted in work of Bacry and Nuyts [2], who still worked in four spacetime dimensions and still subject to spatial isotropy. More recently, this classification was extended to arbitrary dimension via deformation theory: [3] (recovering the classification of Bacry and Nuyts), [4] (in higher spacetime dimension) and [5] (in three spacetime dimensions), but always under the assumption of spatial isotropy. Based on these results, the simply-connected spatially isotropic homogeneous spacetimes (i.e., the kinematical Klein geometries) were classified in [6], resulting in a small class of geometries, including lorentzian, galilean and carrollian, which were further studied in [7].

A natural continuation of this programme is to lift the assumption of spatial isotropy and the natural first step is to consider the case of coisotropy-one: where the rotational subalgebra acts on the tangent sphere (i.e., the space of directions) with codimension-one orbits. This is a much richer class of spacetime geometries and their classification, which is substantially harder, is the goal of a series of papers commencing with this one.

We recall that a $(d+2)$-dimensional spatially isotropic homogeneous spacetime admits a transitive action of a spatially isotropic kinematical Lie group $K$. Its Lie algebra $\mathfrak{k}$ is a real Lie algebra of dimension $(d+2)(d+3)/2$ with a Lie subalgebra $\mathfrak{r} \cong\mathfrak{so}(d+1)$ under which the adjoint representation of $\mathfrak{k}$ decomposes into $\operatorname{ad} \oplus2U \oplus R$, where $\operatorname{ad}$, $U \cong\mathbb{R}^{d+1}$ and $R$ are the adjoint, vector and scalar representations of $\mathfrak{r}$, respectively. Such Lie algebras were the subject of the classical papers already mentioned [1, 2] (for $d=2$) as well as the more recent papers [3, 4, 5] which deal with general $d$. The Klein pair corresponding to a spatially isotropic spacetime is $(\mathfrak{k},\mathfrak{h})$, where $\mathfrak{h} \cong\mathfrak{r} \oplus U$ as an $\mathfrak{r}$-module. Among such Klein pairs, the simplest are those corresponding to Minkowski, Galilei and Carroll spacetimes. We collect them in Table 1. The notation in the table is such that $\mathfrak{k}$ is spanned by $J_{ab}$, $B_a$, $P_a$, $H$ and $\mathfrak{h}$ is spanned by $J_{ab}$, $B_a$, with $J_{ab}$ a basis for $\mathfrak{r}$. Geometrically we may interpret $J_{ab}$ as generating rotations, $B_a$ as generating boosts, $P_a$ generating spatial translations and $H$ generating time translations. The table omits the indices and displays the Lie brackets of $\mathfrak{k}$ in an abbreviated – yet unambiguous – notation explained in [6 §2.2]. We omit the Lie brackets common to all kinematical Lie algebras; that is, those that involve the rotational subalgebra $\mathfrak{r}$, which in this abbreviated notation are $[J,J]=J$, $[J,B]=B$, $[J,P]=P$ and $[J,H]=0$.

The minimal breaking of spatial isotropy (where the rotational algebra $\mathfrak{so}(d+1)$ breaks to $\mathfrak{so}(d)$) in these spacetimes will result in a coisotropy-one homogeneous spacetime whose transitive Lie algebra belongs to one of two classes of Lie algebras we now describe. To this end it is convenient to recall a definition due to Morand [8].

This definition encompasses many classes of Lie algebras relevant to the description of spacetimes. For example, $(1,1)$-KLAs are aristotelian Lie algebras, whereas $(1,2)$-KLAs are the original spatially-isotropic kinematical Lie algebras. The $(2,1)$-KLAs are the Lifshitz Lie algebras, classified in [9], whereas $(2,2)$-KLAs are the boost-extended Lifshitz Lie algebras, which are the subject of ongoing work which will be presented in [10]. The following structural result will be shown in that paper.

This theorem is analogous to a theorem proved in [9], which shows that every $(2,1)$-KLA is an extension-by-grading-element of a $(1,1)$-KLA.

Let us now break spatial isotropy in any of the spacetimes in Table 1. We assume that there is a distinguished nonzero spatial vector $\mathbf{u}\in\mathfrak{k}/\mathfrak{h}$, which breaks the rotational symmetry from $\mathfrak{r}$ to the stabiliser of $\mathbf{u}$, which is a subalgebra $\mathfrak{s}=\mathfrak{so}(\mathbf{u}^{\perp})$ isomorphic to $\mathfrak{so}(d)$. Under $\mathfrak{s}$, the vector representation decomposes as $U=\mathbf{u}^{\perp}\oplus\mathbb{R}\mathbf{u}$, where $\mathbf{u}^{\perp}\cong\mathbb{R}^{d}$ is the vector representation of $\mathfrak{s}$. We define $\mathfrak{g}$ to be the normaliser of $\mathbf{u}\cdot P$ in $\mathfrak{k}$ and consider the Klein pair $(\mathfrak{g},\mathfrak{g}\cap\mathfrak{h})$ as a model of a coisotropy-one homogeneous spacetime. For Minkowski and Carroll spacetimes, $\mathfrak{g}\cap\mathfrak{h}$ does not contain the boost $\mathbf{u}\cdot B$ along $\mathbf{u}$, since $[\mathbf{u}\cdot B,\mathbf{u}\cdot P]=|\mathbf{u}|^{2}H$; whereas for Galilei spacetime $\mathfrak{g}\cap\mathfrak{h}$ does include $\mathbf{u}\cdot B$. This gives rise to two classes of Lie algebras $\mathfrak{g}$.

If $\mathfrak{g}$ does not include the boost $\mathbf{u}\cdot B$, then as a representation of $\mathfrak{s}=\mathfrak{so}(\mathbf{u}^{\perp})$, $\mathfrak{g}$ decomposes as $\mathfrak{g}=\mathfrak{s}\oplus2\mathbf{u}^{\perp}\oplus2\mathbb{R}$, where the two scalars are the translation generators along time and along $\mathbf{u}$. Therefore $\mathfrak{g}$ is a $(2,2)$-KLA.

If $\mathfrak{g}$ does include the boost $\mathbf{u}\cdot B$, then it has an additional $\mathfrak{so}(\mathbf{u}^{\perp})$ scalar, namely $\mathbf{u}\cdot B$ itself. In that case, $\mathfrak{g}=\mathfrak{s}\oplus2\mathbf{u}^{\perp}\oplus3\mathbb{R}$ as a representation of $\mathfrak{s}$ and hence it is a $(3,2)$-KLA, a class of Lie algebras whose classification is the main result of this paper (for $d\ge3$) and a following paper (for $d=2$). A further paper will contain the classification of the corresponding simply-connected coisotropy-one homogeneous spacetimes; that is, the corresponding spacetime Klein geometries.

Let us now summarise the paper and explain its organisation. In § 2 we prove several structure theorems for $(3,2)$-KLAs. We consider $d\ge3$. For $d=1$ there is no distinction between the scalar and vector representations, so every finite-dimensional real Lie algebra is a $(3,2)$-KLA. Such Lie algebras were classified up to isomorphism in [11] and we shall have nothing more to say about them. For $d=2$ there is no distinction between the scalar and adjoint representations and this, together with the fact that the vector representation admits a larger endomorphism ring than when $d\ge3$, makes the classification problem substantially more complicated. In order to keep the papers to a reasonable size, we will present the $d=2$ classification in a separate paper.

In this first paper, our treatment breaks naturally into two cases and this will be reflected throughout: the generic case ($d>3$) and the special case $d=3$. The main results in § 2 are contained in Theorems A (for $d>3$) and B (for $d=3$).

For all $X \in\mathfrak{so}(V)$, $v,v_i \in V$, $w,w_i \in W$ and $B \in\mathfrak{b}$, the Lie algebra structure is given by the brackets

where $\alpha: W \times W \to\mathbb{R}$ is the unique $\mathfrak{b}$-invariant symmetric bilinear form given by