Gerard Thompson

The holonomy Lie algebras of neutral metrics in dimension four

The Lie algebra isomorphism between su(1,1)×su(1,1) and o(2,2) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously known results, furnish a classification of such Lie subalgebras of o(2,2), with only one case remaining unresolved.

Variational connections on Lie groups

Abstract The inverse problem of Lagrangian dynamics is solved for the geodesic spray associated to the canonical symmetric linear connection on a Lie group of dimension three or less. The degree of generality is obtained in each case and concrete Lagrangians are written down.

The inverse problem for six-dimensional codimension two nilradical lie algebras

Ado’s theorem asserts that every real Lie Algebra g of dimension n has a faithful representation as a subalgebra of gl(p,R) for some p. The theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n. This article is concerned with finding representations for a certain class of six-dimensional Lie algebras, specifically, real, indecomposable algebras that have a codimension two nilradical. These algebras were classified by Turkowski and comprise of 40 cases, some of which contain up to four parameters. Linear representations are found for each algebra in these classes: More precisely, a matrix Lie group is given whose Lie algebra corresponds to each algebra in Turkowski’s list and can be found by differentiating and evaluating at the identity element of the group. In addition a basis for the right-invariant vector fields that are dual to the Maurer-Cartan forms are given thereby providing an effective realization of Lie’s third theorem. T...

Normal form of a metric admitting a parallel field of planes

Local coordinate normal forms for two classes of metrics are obtained. The first class is comprised of those metrics that have a field of degenerate, but not necessarily null, planes, thereby extending a result of A. G. Walker. The second class of metrics are Artinian metrics that possess a pair of parallel, complementary, null distributions. This latter class is shown to be equivalent to a certain class of symplectic forms.

Bi-invariant and noninvariant metrics on Lie groups

A restricted version of the inverse problem of Lagrangian dynamics for the canonical linear connection on a Lie group is studied. Specifically for solvable Lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo-Riemannian metric on the corresponding linear Lie group are found. Of the 19 such metrics four are bi-invariant. The Lie algebras are taken from tables compiled originally by Mubarakzyanov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 104–116 (1963)] and Morozov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 161–171 (1958)].

Minimal representations of Lie algebras with non-trivial Levi decomposition

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.

Classifying Quadratic Forms Over in Three Variables

The quadratic forms in three variables over the field are classified. Some remarks are made about the group of equivalences of the quadratic forms.

Sectional and Ricci Curvature for Three-Dimensional Lie Groups

Formulas for the Riemann and Ricci curvature tensors of an invariant metric on a Lie group are determined. The results are applied to a systematic study of the curvature properties of invariant metrics on three-dimensional Lie groups. In each case the metric is reduced by using the automorphism group of the associated Lie algebra. In particular, the maximum and minimum values of the sectional curvature function are determined.

Nonsolvable Subalgebras of

All the simple and then semisimple subalgebras of are found. Each such semisimple subalgebra acts by commutator on . In each case the invariant subspaces are found and the results are used to determine all possible subalgebras of that are not solvable.