Representations of finitary Lie algebras (Q1858241)

For \(F\) an algebraically closed field of characteristic \(\not= 2, 3\) and \(W\) an infinite-dimensional \(F\)-vector space a Lie subalgebra \(L\) of \(\mathfrak{gl}(W)\) all of whose elements have finite rank is said to be finnitary. Finitary Lie algebras \(L\) with \(\text{nil}_W L = (0)\), where \(\text{nil}_W L\) is the maximal ideal of \(L\) consisting of nilpotent transformations, are considered. In such an algebra every ideal contains a minimal ideal \(S\) of \(L\). It is shown that if \(S\) is infinite-dimensional and not nil, then \(SW\) is the finite direct sum of \(L\)-modules, each of which is a nontrivial irreducible \(S\)-module. A classification theorem is also obtained for irreducible Lie subalgebras of \(\mathfrak{gl}(W)\) which contain a nonzero transformation of finite rank. The existence of a local system of finite-dimensional subalgebras of irreducible finitary Lie algebras with certain restrictive properties is a basic tool in this work.