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

### Abstract

Ado’s theorem asserts that every real Lie Algebra of dimension has a faithful representation as a subalgebra of for some . The theorem offers no practical information about the size of in relation to and in principle may be very large compared to . 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 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.

© 2006 American Institute of Physics

Received 08 August 2006
Accepted 05 October 2006
Published online 15 November 2006

Article outline:

I. INTRODUCTION
II. SIX-DIMENSIONAL LIE ALGEBRAS AND TURKOWSKI’S CLASSIFICATION
III. REPRESENTATIONS FOR TURKOWSKI’S ALGEBRAS
IV. THE INVERSE PROBLEM FOR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATION (ODE) SYSTEMS
V. THE INVERSE PROBLEM FOR THE CANONICAL LIE GROUP CONNECTION
VI. REPRESENTATIONS AND LAGRANGIANS

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2006-11-15

2016-09-28

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