International Journal of Algebra, Vol. 3, 2009, no. 18, 873 - 879 A Note on the Decomposition of the Inverse Limit of Finite Dimensional Lie Algebras Nadine J. Ghandour Math. Department Lebanese International University, Lebanon nadine.ghandour@liu.edu.lb Abstract We extend the Levi-Malcev decomposition theorem of finite dimen- sional Lie algebras to the case of profinite dimensional Lie algebras. We prove that the inverse limit L of a surjective inverse system of finite dimensional Lie algebras L = lim ←- L i ,i ∈ I , where I is a countable set with directed partial ordering, can be written as L = R ⊕ S , where R is a prosolvable ideal of L and S is a prosemisimple Lie subalgebra. Mathematics Subject Classification: 14L, 16W, 17B45 Keywords: Inverse limits, Lie algebras 1 Introduction Most of the general theory on Lie algebras has been established for finite dimensional Lie algebras. However, little is known about the general theory of infinite dimensional Lie algebras. An important class of such Lie algebras are the profinite dimensional Lie algebras L = lim ←- L i which are inverse limits of finite dimensional Lie algebras. Such Lie algebras appear as the Lie algebras of proaffine algebraic groups which play an important role in the representation theory of Lie groups. For more detailed information, the reader is reffered to [3], [5], [6], [7], [8]. So it is of interest to extend the basic theory (found for example in [1], [2], [4] ) concerning finite dimensional Lie algebras to the category of profinite dimensional Lie algebras. In this paper, we generalize the Levi-Malcev decomposition theorem of fi- nite dimensional Lie algebras to the case of profinite dimensional Lie algebras L = lim ←- L i ,i ∈ I , where I is a countable set with directed partial ordering. We prove first that the inverse limit L = lim ←- L n (n ∈ N) of a surjective inverse