Four-dimensional Lie algebras

In mathematical general relativity it is common to study solutions of the Einstein equations with symmetry. In other words, solutions are considered which are invariant under the action of a Lie group . (In what follows I will restrict consideration to the vacuum case to avoid having to talk about matter. So a solution means a Lorentzian metric satisfying .) It is usual to concentrate on the four-dimensional case, corresponding to the fact that in everyday life we encounter one time and three space dimensions. One type of solutions with symmetry are the spatially homogeneous ones where the orbits of the group action are three-dimensional and spacelike. Then the Einstein equations reduce from partial differential equations to ordinary differential equations. This is a huge simplification although the solutions of the ODEs obtained are pretty complicated. Here I will make the further assumptions that the Lie group is of dimension three and that it is simply connected. The first of these assumptions is a real restriction but the second is not from my point of view since it does not change the dynamics of the solutions, which is what I am mainly interested in. With these assumptions the unknown can naturally be considered as a one-parameter family of left-invariant Riemannian metrics on a three-dimensional Lie group. These Riemannian metrics are obtained as the metrics induced by the spacetime metric on the orbits of the group action. Any connected three-dimensional Lie group can occur. Connected and simply connected Lie groups are in one to one correspondence with their Lie algebras.Thus it is important to understand what three-dimensional Lie algebras there are. Fortunately there exists a classification which was found by Bianchi in 1898. People working in general relativity call the spatially homogenous solutions of the Einstein equations with symmetry property defined by Lie groups in this way Bianchi models. They use the terminology of Bianchi, who distinguished types I-IX. A lot of work has been done on the dynamics of these solutions. Some more information on this can be found in a previous post on the Mixmaster model.

For reasons of pure mathematical curiosity, or otherwise, it is interesting to ask what happens to all this in space dimensions greater than three. Recently Arne Gödeke has written a diploma thesis on some aspects of this question under my supervision and this has led me to go into the issue in some depth. One thing which naturally comes up is the question of classifying Lie algebras in dimensions. As far as I can see there is not a useful complete classification in general dimensions but there is quite a bit of information available in low dimensions. Here I will concentrate on the case of four dimensions. In that case there is a classification which was found by Fubini in 1904 and since then other people have produced other versions. Having worked with Bianchi models for many years I feel very much at home with the three-dimensional Lie algebras. In contrast the four-dimensional classification appeared to me quite inhospitable and so I have invested some time in trying to fit the four-dimensional Lie algebras into a framework which I find more appealing. I record some of what I found here. The best guide I found was the work of Sigbjørn Hervik, in particular his paper in Class. Quantum Grav. 19, 5409 (cf. arXiv:gr-qc/0207079).

From the point of view of the dynamics of the Einstein equations one Bianchi type which is notably different from all others is type IX.The reason for this is that the Lie group (which is ) admits left-invariant metrics of positive scalar curvature. Is there a natural analogue for four-dimensional Lie algebras? A useful tool here is the Levi-Malcev theorem which provides a way of splitting a general Lie algebra into two simpler pieces. More precisely it says that each Lie algebra is the semidirect sum of a semisimple and a solvable Lie algebra. The semisimple part is called a Levi subalgebra and is unique up to isomorphism. It turns out that the information about whether there exists a metric of positive scalar curvature is contained in the Levi subalgebra. There are not many semisimple Lie algebras in low dimensions. In fact in dimension no greater than four there are only two, and . These correspond to Bianchi types IX and VIII respectively. The only possible non-trivial Levi decompositions are the semidirect sum of one of the two Lie algebras just mentioned and the real numbers. In fact it turns out that the semidirect sum of a semisimple Lie algebra and the real numbers is automatically a direct sum because any derivation of a semisimple Lie algebra is an inner derivation. The corresponding simply connected Lie group is a direct product. It can be concluded from this that the only simply connected and connected four-dimensional Lie group which admits a metric of positive scalar curvature is . This is the analogue of Bianchi type IX for .

It is common in general relativity to divide the three-dimensional Lie algebras into two disjoint classes, Class A and Class B. The first of these consist of the unimodular Lie algebras, i.e. those whose structure constants have vanishing trace. They are closely associated with the class of Lie groups whose left-invariant metrics can be compactified by taking the quotient by a discrete group of isometries. They also have the pleasant property that their dynamics can be reduced to the case where the matrix of components of the metric in a suitable basis of left-invariant one-forms is diagonal. This is important for the Wainwright-Hsu system, a dynamical system formulation of the Einstein equations for Class A Bianchi models which is the basis for most of the rigorous results on the dynamics of these solutions obtained up to now. If type IX is omitted there are five different Lie algebras in Class A. One way of getting unimodular Lie algebras of dimension four is to take the direct sum of the three dimensional Lie algebras with the real numbers. Call the others indecomposable. The indecomposable unimodular four-dimensional Lie algebras can be classified into six types. Four of these are individual Lie algebras while the other types are one-parameter families of non-isomorphic algebras. One way of putting these into a larger framework is to note that each of them has a three-dimensional Abelian subalgebra. They can therefore be considered as special cases of solutions with three commuting spacelike Killing vector fields. This generalizes the fact that all the Class A Bianchi types except VIII and IX can be considered as solutions with two commuting Killing vector fields. I do not have an overview of the questions of compactification and diagonalization for these metrics. It seems that calculations done by Isenberg, Jackson and Lu in their study of the Ricci flow on homogeneous four-dimensional manifolds (Commun. Anal. Geom. 14, 345) might be helpful in this context.

More details on some of the things mentioned in this post will be given in a forthcoming preprint by Gödeke and myself.

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