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This article points out some of the shortcomings of a paper by Ceballos et al. In particular, representations of the indecomposable six‐dimensional complex nilpotent Lie algebras are corrected. In addition, the class of representations is extended so as to include all the indecomposable six‐dimensional real nilpotent Lie algebras. A test case is studied in detail in which it is shown that there is no representation of one of the six‐dimensional real nilpotent Lie algebras using 5 × 5 matrices. This test case points up some of the severe difficulties that one will encounter when implementing an algorithm such as the one developed by Ceballos et al. The case of the seven‐dimensional complex nilpotent Lie algebras is briefly touched upon and several representations are corrected. [ABSTRACT FROM AUTHOR]

For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of the associated canonical Lie group connection are given. In each case, a basis for the associated Lie algebra of symmetries is constructed and analyzed. [ABSTRACT FROM AUTHOR]

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Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the case of solvable Lie algebras that have a codimension one Abelian nilradical. It is argued that such algebras are associated to Einstein metrics only when the metric is of constant negative or zero curvature. Thereafter, all of the four- dimensional Lie algebras are examined with regard to the question of whether they correspond to Einstein metrics. For algebras that are not of the type solvable with codimension one Abelian nilradical, the space of derivations is found and used to reduce the metric and the Ricci tensor is calculated. It is then possible to say precisely which metrics are Einstein spaces. In the indecomposable case precisely four such Einstein spaces are obtained. Two of them are spaces of constant curvature. No examples are found in the decomposable case. [ABSTRACT FROM AUTHOR]

Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the problem of characterizing invariant metrics of zero and non-zero constant curvature. Killing vector fields for such metrics are constructed and play an important role in the case of at metrics. [ABSTRACT FROM AUTHOR]

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized. [ABSTRACT FROM AUTHOR]

We obtain a matrix representation for each of the indecomposable 9-dimensional real Lie algebras that have a non-trivial Levi decomposition. [ABSTRACT FROM AUTHOR]