Research Article Nonsolvable Subalgebras of gl(4, R) Ryad Ghanam 1 and Gerard Thompson 2 1 Department of Mathematics, Virginia Commonwealth University in Qatar, P.O. Box 8095, Doha, Qatar 2 Department of Mathematics, University of Toledo, Toledo, OH 43606, USA Correspondence should be addressed to Ryad Ghanam; raghanam@vcu.edu Received 17 June 2016; Accepted 26 July 2016 Academic Editor: Rutwig Campoamor-Stursberg Copyright © 2016 R. Ghanam and G. Tompson. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All the simple and then semisimple subalgebras of gl(4, R) are found. Each such semisimple subalgebra acts by commutator on gl(4, R). In each case the invariant subspaces are found and the results are used to determine all possible subalgebras of gl(4, R) that are not solvable. 1. Introduction In this paper we consider the problem of classifying the Lie subalgebras of gl(4, R). In Section 2 we will explain in what sense we classify the Lie subalgebras. It is a well known fact in Lie theory that every Lie algebra g admits a semidirect Levi decomposition; that is to say, g ≈⋊, where  is a semisimple subalgebra and  is a solvable ideal. We will divide the project into two parts. First of all we will fnd all the Levi algebras where  is nonzero. In a separate venue we will examine the solvable algebras. Details about the subalgebras of gl(2, R) and gl(3, R) may be found in [1]. One could of course opt to classify the Lie subalgebras of sl(4, R) instead of gl(4, R), which has the advantage of avoiding many trivial and obvious subalgebras; however, the drawback is that in many other cases one has to impose a somewhat arbitrary condition on the parameters in order to obtain a subalgebra of sl(4, R) rather than gl(4, R) so we have opted to stick with the latter. Every abstract semisimple subalgebra is a direct sum of simple subalgebras. By “abstract” here we mean that the given subalgebra does not necessarily appear as a subalgebra of gl(, R) for some , although Ado’s theorem informs us that such  must exist. For small values of  it is feasible to fnd all the irreducible representations of such a simple subalgebra in gl(, R) and hence, up to isomorphism, all the representations of the semisimple subalgebra, which are direct sums of irreducible representations. Suppose that we start with a certain semisimple sub- algebra ⊂ gl(, R). We will work with that particular representation and not change the semisimple part of the algebra until we have found all possible subalgebras that have  in that representation as its semisimple part. Ten  acts on gl(, R) as a Lie algebra by () = [,]; that is, commutator defnes a Lie algebra homomorphism from  into End(gl(, R)), as readily follows from the Jacobi identity. Since  is semisimple, this representation must be completely reducible. To construct Levi subalgebras of gl(, R) we have to fnd the irreducible submodules or invariant subspaces of this representation. A particular submodule may or may not defne a solvable subalgebra of gl(, R). In order to fnd all possible Levi subalgebras corresponding to a particular  we have to see if it is possible to add any of the submodules together so as to defne a larger solvable subalgebra. However, it is not true that every submodule is a sum of the basic submodules; see, for example, the six-dimensional subalgebra sl(2, R)⋊ 3.1 where two submodules are “correlated.” Tus we have the following four-step procedure: (i) Find all simple subalgebras of gl(, R) for the given value of . (ii) Find all possible representations (up to change of basis) of these simple subalgebras of gl(, R). (iii) Find all semisimple (not simple) subalgebras of gl(, R) and their representations. Hindawi Publishing Corporation Journal of Mathematics Volume 2016, Article ID 2570147, 17 pages http://dx.doi.org/10.1155/2016/2570147