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. The geodesic spray of the canonical symmetric connection for each of the 40 linear Lie group G is given. Thereafter the inverse problem of the calculus of variations for each of the geodesic sprays is investigated. In all cases it is determined whether the spray is of Euler-Lagrange type and in the affirmative case at least one concrete Lagrangian is written down. In none of the cases is there a Lagrangian of metric type. (c) 2006 American Institute of Physics.