On the Classification of the Real Four-Dimensional Lie Algebras

Extremal horizons with reduced symmetry: hyperscaling violation, stripes, and a classification for the homogeneous case

“…This makes it clear that the relevant symmetry structure could more generally involve real fourdimensional Lie algebras with a preferred three sub-algebra (which, roughly, generates the spatial symmetry group in the dual field theory). 2 Happily, the classification of such algebras (with the relevant subalgebras) has also been accomplished, though much more recently -see [44] for a clear exposition and also [45] for the proper history. Here, we use these results to find the more general classification of extremal black brane geometries relevant for holography in 5 bulk dimensions.…”

“…It should be clear in each case that in providing examples, we have barely scratched the surface of what are likely very rich sets of solutions to the Einstein equations or appropriate low-energy limits of string theory. In the direction of finding a more complete classification, we discussed the possible application of the larger algebraic structures uncovered in [44,45] to classify 5d extremal near-horizon geometries in terms of real four-algebras with preferred 3d subgroups. While we found that several of these possibilities will remain unrealized in sensible gravity coupled to matter theories (which satisfy the Null Energy Condition), others can be realized with simple matter sectors coupled to gravity, and likely arise as duals to suitable infrared phases in strongly-coupled quantum field theory.…”

Ricci flow on locally homogeneous closed 4-manifolds

“…According to the classification of the 4-dimensional unimodular Lie algebras ( [13]), for each such algebra, there is some basis X 1 , X 2 , X 3 , X 4 such that the Lie bracket takes the form indicated below. We adopt the notation 1 in [13].…”

Control affine systems on solvable three-dimensional Lie groups, I

“…The classification over C was done by S. Lie (1893), whereas the standard enumeration of the real cases is that of L. Bianchi (1918). In more recent times, a different (method of) classification was introduced by C. Behr (1968) and others (see [14], [13], [15] and the references therein); this is customarily referred to as the Bianchi-Behr classification (or even the "Bianchi-Schücking-Behr classification"). Any solvable three-dimensional Lie algebra is isomorphic to one of nine types (in fact, there are seven algebras and two parametrised infinite families of algebras).…”